Like there is no difference in Vishnu, Krishna, Raama, NriSimha and other manifestation. But sometime Bhairavanatha appears a little different from Lord Shiva, as if He is a part incarnation of Lord Shiva. Is Bhairavanatha completely same as Shiva or is He a part incarnate?
Lord Shiva took his fifth incarnation as Bhairav and his Shakti was called 'Bhairavi'. So he is incarnation of shiva .
An incarnation may not be same as the one who is taking avatar like narsimha is not like how lord Vishnu is shown.
Shiva Purana says in
1.1.7 Shiva Obliges Brahma Nandikeshwar continuing with the tale said: "After according same status to Vishnuji as that of his own, Lord Shiva opened his third eye from it manifested 'Bhairav'. He ordered Bhairav to kill Brahmaji. Bhairav severed the fifth head of Lord Brahma with his sword. Brahmaji became very terrified. He was trembling in fear. Lord Vishnu felt pity on his condition and requested Lord Shiva to forgive him.
1.3.16 The first incarnation of Lord Shiva was as Mahakal and his corresponding Shakti was called Mahakali. His second incarnation is Tar and his Shakti was called ‘Tara’. The third incarnation is Bhuvaneshwar and his Shakti was called ‘Bhuvaneshwari’. Fourth incarnation as ‘Shodash’ and his Shakti was called ‘Shodashi’ or ‘Shri’. Fifth incarnation as Bhairav and his Shakti was called ‘Bhairavi’. Sixth incarnation of Lord Shiva is famous as ‘Chhinamastak’ and his Shakti by the name of ‘Chhinamasta. Seventh incarnation as ‘Dhoomvan’ and his Shakti was known as ‘Dhoomvati. Eighth incarnation was as Baglamukh and his Shakti as Baglamukhi. Nineth incarnation of lord Shiva became famous as Matang and his Shakti as ‘Matangi’. Tenth incarnation as ‘Kamal’ and his Shakti as Kamala.
by the way while searching for Partial incarnation I could only find Ashwathama as partial Incarnation which is mentioned here
1.3.29 Indra went to Arjuna in the guise of a celibate and asked about the purpose for which he was doing penance. Arjuna told him that he wanted to defeat the Kauravas. Indra then told Arjuna that it was not in his capacity to help him achieve victory over the Kauravas, because of Aswatthama who was a partial incarnation of lord Shiva.
Chapter 8 says
भैरवः पूर्वरूपो हि शङ्करस्य परात्मनः।
मूढास्ते वै न जानन्ति मोहिताः शिवमायया॥२॥
Bhairava happens to be the complete form of Siva. The people overpowered with the illusion of Siva, remain ignorant about the Same.
It says Bhairava is full form of Shiva Paramatma.
Shiva created Bhairava as part of sports or leela, to behead creator Brahma (To know more about leela, refer this answer).
प्राक्च पङ्कजजन्मासौ शास्यस्ते कालभैरव!।
कालवद्राजसे साक्षात्कालराजस्ततो भवान्॥४६॥
Siva said, “O Kalabhairava, initially, you punish this Brahma. You would possess the lustre of the god of death and shall be known as Kalaraja.
विश्वं भर्तुं समर्थोऽसि भीषणाद्धैरवः स्मृतः।
त्वत्तो भेष्यति कालोऽपि ततस्त्वं कालभैरवः॥ ४७॥
You are known as Bhairava because you possess horrible features and are capable of supporting the earth. Because even Kala is afraid of you, therefore, you are known as Bhairava. Even Kala will be afraid of you. Therefore you would be known as Kālabhairava.
Shiva also created Bramahatya as a part of leela and said it to inflict Bhairava and told him to get liberate from it in holy city of Kasi or Varanasi. Bhairava took Vrata of Brahmahatya at command of Lord Shiva.
वामाङ्गुलिनखाग्रेण चकर्त्त च विधेः शिर:॥५२॥
Nandisvara said, "Accepting these boons, Kalabhairava, atonce, severed the fifth head of Brahma, with the nail of his left finger.
इत्युक्त्वा पश्यतस्तस्य तेजोरूपः शिवोऽब्रवीत्।
उत्पाद्य चैकां कन्यान्तु ब्रह्महत्याभिविश्रुताम्॥ ६३॥
Thus speaking and in full view of all, Siva produced a small girl named Brahmahatya. Then Siva in the form of mass flame spoke.
तावत्त्वं भीषणं कालमनुगच्छोग्ररूपिणम्॥६४॥
“Until she goes to the divine city of Varanasi, you too shall follow her, for a long period.
सर्वत्र ते प्रवेशोऽस्ति त्यक्त्वा वाराणसीं पुरीम्।
वाराणसीं यदा गच्छेत्तन्मुक्ता भव तत्क्षणात्॥६५॥
You can freely move to any place, except the city of Varanasi. On reaching Varanasi, you would be freed from Brahmahatya.
In next chapter (Chapter 9), Bhairava is praised as full Shiva by Lord Vishnu. Verse 6 says Bhairava has amsa of Shiva and also says Bhairava has full form like Shiva.
महादेवांशसम्भूतं पूर्णाकारं च भैरवम्॥ ६॥
mahādevāṃśasambhūtaṃ pūrṇākāraṃ ca bhairavam॥ 6॥
Bhairava is born from amsa of Mahadeva and also has full form like Shiva.
When Bhairava goes to Narayana for begging alms due to Brahmahatya sin (as part of Shiva's leela), Narayana praises Bhairava as follows:
अयं धाता विधाता च लोकानां प्रभुरीश्वरः।
अनादिः शरणं शान्तः पुरः घड्वंशसम्मितः॥ ११॥
He is the creator the preserver, besides being the lord of the world. He is the primordial being, the lord and the refuge of all. He is calm and greater than the twenty six principles.
सर्वज्ञः सर्वयोगीशः सर्वभूतैकनायकः।
सर्वभूतान्तरात्माऽयं सर्वेषां सर्वदः सदा।। १२॥
He is the omniscient, lord of the yogis, the sole leader of all the beings, the internal soul of all the living beings and he always bestows everything to everyone.
ये विनिद्रा विनिश्वासाः शान्ता ध्यानपरायणाः।
धिया पश्यन्ति हृदये सोऽयं पद्मे ! समीक्षताम्।। १३।।
O Laksmi, look, he is the lord whom the conscious people, are engaged in yogic practices, visualise in their hearts, discarding their sleep, stopping their breathing, concentrating their minds. He happens to be the same. You better visualise him with the vision of your knowledge, in your own heart.
यं विदुर्वेदतत्त्वज्ञा योगिनो यतमानसाः।
अरूपो रूपवान्भूत्वा सोऽयमायाति सर्वगः।। १४॥
He is the one who is known to the yogis who are well-versed in the Vedic tattvas. He is the one, who being formless, is at times visible in some definite form. The one who is all pervading is arriving here.
तं दृष्ट्रा न पुनर्जन्म लभ्यते मानवैर्भुवि।
सोऽयमायाति भगवांस्त्र्यम्बकः शशिभूषणः॥ १६॥
At the mere sight of the lord, the people are never reborn on earth. Such a type of lord Tryambaka who has the crescent as an ornament, is arriving here.
पुण्डरीकदलायामे धन्ये मेऽद्य विलोचने।
यद्दृश्यते महादेवो ह्याभ्यां लक्ष्मि ! महेश्वरः॥ १७॥
O Laksmi, my eyes resembling a lotus leaf, have become graceful today because I would have an opportunity to have a look with these eyes, Mahesvara, (Maheshwara) the great lord.
धिग्थिक्यदन्तु देवानां परं दृष्टा न शङ्करम्।
लभ्यते यत्र निर्वाणं सर्वदुःखान्तकृतुि यत्।। १८ ।।
Disgrace to the position of the gods, who have not seen lord Shankara (Siva) so far, who bestows salvation, and who is the remover of all the miseries.
देवात्वादशुभ किञ्चिद्देवलोके न विद्यते।
दृष्ट्राऽपि सर्वे देवेशं यन्मुक्तिं न लभामहे॥ १९॥
What is use of becoming a god in the heaven, in case, we people are unable to achieve salvation by looking at Mahadeva.”
You can see Lord Shiva's epithets are used for Bhairava. So, Lord Bhairava is verily Lord Shiva Himself. You can read complete chapter to know about Vishnu's obeisances to Bhairava.
Even Adi Shankara's Kalabhairava Astkakam describes Bhairava as complete Shiva.
As a side note, Tantras like Vijnana Bhairava Tantra descibe Bhairava as Supreme Lord (with profound philosophy).
Here I explain Bhairava philosophically according to īśvarādvayavāda (Trika philosophy) According to it, Bhairava is the ultimate reality beyond the contemplation, it is fully transcendental. On questions asked by Devī, Lord Shiva describes Bhairava as follows.
Good! Good! Dear one, you have put questions which pertain to the very quintessence of Tantra. Though the matter is most esoteric, oh auspicious one, yet shall I explain it to you. Whatever has been declared to be the composite form (Sakala¹) of Bhairava, that oh goddess should be considered as insubstantial (asāratayā), as phantasmagoria (lit. as the net of Indra), as magical illusion (māyā), as dream, as the mirage of a town of Gandharva in the sky. The sakal aspect of Bhairava is taught, as a prop for meditation, to those who are of deluded intellect, who are interested in ostentatious performance of rituals, it has been declared for those people who are a prey to dichotomising thought constructs. (Verse 7-10, Vijnānabhairav tantra)
1- All manifestation from gods down to the mineral is known as sakala. Sakala is the sphere of Māyā tattva. It con-sists of bheda-difference or division. The essential nature of Bhairava cannot be known by means of sakala which consists of difference and division.
Then he describes the real nature of Bhairava as follows-
In reality, Bhairava is neither of the form of nine (navātma), nor a multitude of letters (sabdarāsi) nor of the three heads (trishirā) nor of three Sakti, nor consisting of nāda and bindu, nor of ardhacandra, nirodhikā, etc., nor is His essence concerned with the piercing of the (six) chakras, nor does sakti or Energy constitute His essence.
The above concepts are used for those whose intellect is not yet mature enough to grasp Reality (in its highest aspect), Just as a bogy is used to frighten away children from their obstinacy for getting some worthless or undesirable thing. These concepts play the same role as the bonbon of the mother. They are meant to induce the aspirants to tread the path of righteousness and spiritual practices in order that they may ultimately realize the nature of Bhairava which is non-different from their essential Self.
The hermeneutic etymology of the word Bhairava has been given in various ways in Trika philosophy. The important interpretations are given below.
1- The word 'Bhairava' is composed of four letters- 'bhā' + 'ai' + 'ra' + 'va'. 'bhā' + 'ai' by the rule of sandhi becomes bhai. This bhai together with 'ra', 'va' becomes bhairava. Each of these letters connotes certain important ideas. Bhā is a word by itself which means 'light' i.e. the light of consciousness in this context; ai according to Trika philosophy is symbolic of kriyāshakti (the power of activity); rava connotes ravayati which, in this context, means vimarshati i.e. comprehends. So according to this etymology Bhairava means "He whose light of consciousness-joined with his ·power of activity ('bhā' + ai i.e. bhai) comprehends (ravayati) the entire universe is His Self".
2- In Parātrimshikā, Abhinavgupta gives another interpretation of Bhairava viz.,'
Bhairavo bharañātmako mahāmantraravātmakascha" (p.63).
i.e. 'Bhairava is one who supports and protects the cosmos and (constantly) sounds the great mantra of 'I' (aham).
3- He offers intrepidity to those who are terrified by the cycle of transmigratory existence.
4- Each letter of Bhairava is symbolic of His three main activities; Bha is symbolic of bharaña or maintenance of the universe; ra is symbolic of ravaña or withdrawal of the universe; va is symbolic of vamana or projection of the universe.
The two adjectives of 'Bhairava' viz., sarvadah (bestower of all) and vyāpakah (all-pervasive) are also connected by implication with two letters of Bhairava, viz., ra and va; ra is symbolic of the root ra which means to give, to grant, to bestow. He is called sarvadah, because he bestows everything, 'va' is symbolic of va which means 'to be diffused' Bhairava is called vyāpakah because of the diffusion of His presence everywhere.
So, as a conclusion Bhairava if taken with respect to non-māyā sphere, it is the ultimate reality with more emphasis on Shiva part with his Shakti. (According to Trika the ultimate reality is Shiva along with Shakti as inseparable).
I am also deeply perplexed by this question. There is one Supreme Bhairava (the one who is speaking the Vijyana Bhairva Tantram for example and several others, and there he is obviously the Supreme Brahman).
Then, we have Kaala Bhairava and we also have the many Batuka Bhairavas.
There is one Swarnaakarashana Bhairava stotram found in Rudra Yamala. The very first verse of the stotram is:
om namaste bhairavaaya brahma vishNu shivaatmane
Now, in this verse, Lord Bhairava is described as the sarvagata paramatman who is antaryami (or the inner self) of even the Trinities. So clear distinction has been made between him and Lord Shiva and in other way round.
So, IMO, there are subtle and mysterious differences between Lord Shiva and Lord Bhairava which can be known only to the highest level Sadhakas. The Bhairva Tattva, i think, is intentionally kept secret in all scriptures.
That's why and strangely we don't have much on Lord Bhairava either except for the above stotram and another stotram composed by the great Kashmiri Acharaya Abhinava Gupta which one can read here.