Few Basics: In the Rig Vedā, the word "Śiva" occurs more than 30 times, and all these are not personified names (swarūpa), they are various aspects of auspiciousness and pleasantness. For example, the hymn "Be Śiva with me" meaning be gentle with me. The first homage to Rudra calls Him pleasant or peaceful, then all the synonymous titles follow - such as benevolent, caring, prosperous, glowing, gentle, knowledge & eloquence, love, bond, friendly, pure, generous, compassionate, pure/pious, and many more. In this way Śiva: the auspiciousness pervades forever, hence the title “Sadaa:Śivam”. As the famous Tamil Poet Thirumular, composer of Tirumurai says, Anbe Sivam (அன்பு சிவம்), meaning "Compassion/affection itself is Sivam". In this, the all-pervasive entirety is Viṣṇu/Hari who manifests beyond kālá (time). Viṣṇu is the canvas for creation, whereas the creative idea is called Brahmā. In this creation manifest countless beings with a mind, this mind becomes the ruler - which is Indra. Sri Aurobindo says, "the invisible force that propels creation from the lowest to the highest state of Viṣṇu is called Rudra"
Pṛśni is also a name of the wife of Prajāpati, which is also correct as Prajāpati is the Yajamana (the one performing the Yajñá). Pṛśni is also a name of a wife of Saptarishi (the 7 primordial divine sages). Maruts are also said to be the children of Rodasī (consort of Rudra) and the same hymn distinctly addresses Rudra as their father RV5.57. Rudra is not necessarily a single entity or a group because of the unique nature of Rudra to manifest across various concepts and divinities. Though Rodasī was mentioned, a single entity called Śiva as a single Rudra is yet to be introduced in our discussion. Since Rudra is the raw and un-manifested invisible force, with the knowledge and immortal wisdom came the Maruts as the individual forces of nature wielded by Rudra. These forces cause the evolution of Creation. They can save us and, at the same time, cause destruction to restructure things in Creation. Hence, the Maruts are called the arrows/missiles of Rudra, which we will see in detail.
Appearance aspect: Ṛṣis of the Vedās were very explicit about the appearance of Rudra and the commonality with that of Śiva. While the first declaration to Rudra is seen in the Rig Vedā, the Saṃhitās of Atharva & Kṛṣṇa Yajur Vedās give utmost detail on His features and nature. To start with, Rig Vedā 1.43 - which is the first declaration to Rudra - says in hymn 5: "He is brilliant like the sun, dazzling like gold, and the best of the divine and of Vasu", similarly Kṛṣṇa Yajur Vedā Taittirīya Saṃhitā 4.5.5 says "Oh lord of mountains, whose brilliance/rays which permeates". Across the Vedās, His form is Viṣvarūpam, meaning the cosmic form or the all-encompassing omni-form RV2.33.10,TS4.5.4,TA10.23.1, or Pururūpam, meaning multiform/multihued.
Empirical View: Atharva Veda Chapter 10 specifies that "The Supreme Being keeps making Soma, the eternal delight" which is empirical to the expression of amṛta dripping from the crescent moon on Śiva's head, but before we make this conclusion let's explore further. Sometimes Soma is the full moon belonging to the Tiṣya/Pusya month, on this day oblations are given to Rudra. Monday is called Somavara and is the day of Śiva TS2.2.10.
A much clearer description is given in Brahmana 1 of Sukla Yajur Vedā Chapter 3, called Śatarudrīya, in an Anuvākam given to Drāpa "the remover/dispeller". So, who is this dispeller? It is Andhasah, meaning Soma, and Rudra is Andhasaspati "अन्धसस्पते"TS4.5.10, meaning the "Lord of Soma". Hence, in the Itihasa and Puráńic realm, Śiva is iconified as Somnath - with a crescent moon on the forehead dripping amṛta. Many derivatives then emerged - like Somashekhara in Peringottukara Kerala, Somadeva, Somaskanda, and even Somavāra (a day of Śiva) and Someshwara in Kolar and Somanath in Gujrat.There are various names given to the three-eyed Siva, like trayambakam and Virūpākṣa, but there exists a celebrated title "Soma:suragni Lochana", meaning the one with Soma, Surya and Agni as His three eyes. This reference can be found in Atharva Vedā 15th Kanda 18th Prayāya, famously known as Vrātyas Suktam, which says "oh Vrātya, as for this right eye is the distant Sun/Āditaya, so for the left eye is the Moon/Soma". But what about the Agni of the 3rd eye which burnt Kamadeva? There is no direct declaration in the Vedas, but Śrī Rudram of the YajurVedā urges Rudra's missiles not to hit us, the same Vedā urges Agni - heat of those missiles - not to harm us, but to be auspicious for us TS 4.61.
Agni: Agni disintegrates everything, from Yajñá to a corpse - which is His aghora (fierce) side - but He remains unblemished, so He is Śiva (Forever Auspicious) and Mīḍhvaḥ (gracious) RV 3.16,7.34,4.6. But Taittirīya Saṃhitā, esp. Śatarudrāya, goes much deeper and clearer than this, by addressing His aghora tanūr (fierce body) as Rudra and the auspicious body as ŚivaTS 2.2, 5.7.3,4.5.1/10. Hence the two famous hymns from Yajur Vedā Taittirīya Saṃhitā – Śrī Rudram “Oṃ namaḥ śivāya” and "या ते रुद्र शिवा तनूः"TS4.5.10 meaning Rudra who has the auspicious/Siva body. So, when approaching our father we prefer his gentle and caring side and not his fierce side.
Contrast: One other Divinity who shares similar attributes, including physical attributes, and who is addressed as the Asura of Devas, and a few very specific titles - like Rudra - is Indra. The Atharva Veda points to Indra's role as the supreme ruler in terms of administration of Creation and Yajñá, but there is no clear reference of him being addressed as one of the Rudras. Many scholars recognized Vayu, to some extent, as Rudra due to the distinct ability to be able to give birth to Maruts and a few acknowledge similarities to īśana (Rudra) RV 1.134. It is because of this ability of manifesting into various divinities, especially in the most fierce aspects of divinities, that Rudra is greatly feared by Adhvaryu, sages, Ṛṣi and other Devatas during the meticulous process of Yajñá. On the other hand, the commoners fear Rudra for their cattle, their families, children, offspring, homes and farms. But, as we discussed, Rudra becomes a polar contradiction, wherein the same sages, Ṛṣi and commoners sing to Rudra and Maruts for their protection, their wellbeing, their health, for knowledge and for liberation from karma/death TS4.5.10. This fierce Rudra is also gentle, auspicious, friendly, a doctor who heals with medicines, gives immortality, deliverer to Yama (the path of Truth), the abode of Truth (rta), provider of wealth VS10.20,3.57, the three-eyed liberator(Triambaka), and protector of descendants, making him the most diverse, independent, foremost (Sriṣṭhaḥ) and contradictory Divinity - who is WHOLE, and not dependent on anyone, but self-supreme (Svadhanva) father to all.
या ते रुद्र शिवा तनूरघो॒रापापकाशिनी ॥ 4.5.1 You (या) Rudra of
Auspicious (शि॒वा) body (तनूः), and of fierce form (अघोरा) is the
dispeller of our darkness and karmic remnants (अ-पापकाशिनी ) ~Yajur Vedā Taittirīya Saṃhitā 4.5.1
So Rudra takes up Auspicious Form/body/manifestation = Siva and Viceversa.
या ते रुद्र शिवा तनूरघोराऽपापकाशिनी।तया नस्तनुवा शंतमया
गिरिशन्ताभिचाकशीहि॥3.5 You (या) Rudra of Auspicious (शिवा) body (तनू),
and of fierce form (अघोरा) resident of highest place (गिरिशन्त) with
your brilliant design/thought (अभिचा +काश्) which removes our avidya
and karmic cycle of pāpa (अ-पापकाशिनी ) make us (नः) realize our
blissful clam self(शंतमया + तनू +तया ) ततः परं ब्रह्मपरं बृहन्तं
यथानिकायं सर्वभूतेषु गूढम्।विश्वस्यैकं परिवेष्टितारमीशं तं
ज्ञात्वाऽमृता भवन्ति॥3.7 Henceforth/moreover, further than (ततः परं)
Greatness known to us/our personal God (बृहन्त) is Brahman/infinite
that is further/superior (ब्रह्म+परं ) thought each enjoyed their own
bodies/individuality (यथानिकायम्) He is the indweller hidden in all
(सर्व+भूतेषु+गूढम् ) he along encompasses all entirety and is its
authority (विश्वस्य+ एकं+ परिवेष्टितारम + ईशाम् ) knowing/realizing
that one becomes amṛta/immortal (ज्ञात्वा+अमृताः ) ~ Svetasvatara
Upaniṣhad 3.5, 3.7
Same is justified in the Upanishad too. So Rudra takes up Auspicious Form/body/manifestation = Siva and Viceversa.
Conclusion: All these aspects match well with the Puráńas that define Śiva and the Ganas that surround Him in thousands and thousands of groups and leagues KYV4.5,VS16.6. They describe him as the one residing on mountains, seated on the highest position, fair in complexion, with locks of hair, wearing deerskin, handsome and muscular, with three eyes and a blue neck, holding a spear (Tri:Shula). We see the divine contradiction that He is both fierce and benevolent/adorable KYV4.5.10, ferocious yet gentle, supreme yet reachable, nourisher yet destroyer, a father to both pious and nefarious, a calm ascetic and a supreme dancer, simple yet dazzling. He is the wave and the calm sea, and many more immeasurable contradictions. As we enter into the Yajur Vedā, especially into Śrī Rudram and Śatarudrīya, aspects of Rudra, Śiva and Soma become crystal clear. The question always remains, why is Rudra of the Vedās addressed as Śiva in the Puráńas? Based on our discussion so far, the answer is quite simple. Whereas the Vedās focused on the cosmic phenomenon called Rta and revolved around Yajñá, the Puráńas and Itihāsa focused on the benevolent side of Rudra - his nourishing, healing, protecting, unconditional and blissful nature - and Śiva (the Auspiciousness One) was most suited.
For complete detail please google "Sanatan Dharma Siva Rudra across Vedas and Itihasa"