Vishnu Purana and Srimad Bhagavatam are two important Vaishnava puranas. Of the two, which is given more importance in Sri, Madhva, Pushti & Gaudiya Vaishnavism? why if any reason?
Sri Vaishnavas, who are worshippers of Lakshmi-Narayana considers the Vishnu Purana as the best of all Puranas due to its primary focus on the supremacy of Sri Vishnu. This is what Sripad Ramanujacharya says about the importance of Vishnu Purana in His "Vedartha Sangraha":
The Vishnu-Purana has been unanimously adopted by all the great sages, from the East, West, North and South, as an all-sufficient authority for the ascertainment of all righteous conduct and Reality. On the authority of the aphorism, ‘Whence the origin etc. of this (B.S. 1:1: 2)’, Brahman is understood to be the cause of creation etc. of the universe. Now, the Purana in question, opens with a clear statement of the question as to what constitutes the source of the creation etc. of the universe and proceeds to answer, ‘From Vishnu is all this born (Vi. Pu. 1:1:31)’. All agree on the point that such a procedure proves that the work has as its sole purpose the exposition of the specific character of Brahman. In the same text we have the statement, ‘She, who has been named Prakriti by me, has both a manifest form and unmanifest form. Both she and the Purusha (individual self) dissolve in the Paramatman. Paramatman is the supporting ground of all and is the supreme godhead. He is sung in the Veda, as well as Vedanta, as Vishnu (Vi. Pu. 6:4: 39)’. In all the Vedas and Upanishads, he is sung as the ultimate cause through all modes of expression.
Even as among all the srutis, the Narayana-anuvaka, has as its sole aim the specific determination of nature of Brahman, Vishnu-Purana is also a work solely devoted to that particular end. It opens with these questions (Vi. Pu. 1:1: 4-5): ‘O Knower of dharma, I desire to hear from you how this world was and how it will be in future. By what is this world filled ? Whence are these moving and non-moving beings? By what is it brought to dissolution and into what does it dissolve?’ The question in essence is ‘What is Brahman?’ The answer making a decisive declaration of the nature of Brahman follows. ‘The universe originates from Vishnu and exists in him. He is its maintainer and controller. He is the world. He is greater than the greatest. He is supreme. He is the Paramatman. He dwells in the selves. He is beyond all characterizations in terms of form, colour, etc. He is beyond decay and cessation, beyond modification, increase and characterizations in terms of forms, colour, etc. He is beyond birth. He can be described only as the ‘Ever-existent’. He dwells in all and all things dwell in him. Therefore the wise name him ‘Vasudeva’. He is the supreme Brahman, eternal, unborn, imperishable and undecaying. He is always of the self-same form, free from evil, and thus pure. He is all this and has both this manifested and unmanifested as his form. And also he exists as the individual spirit and time (Vi. Pu. 2:1:1-14). He transcends, O Sage, the character of all material elements, all mutations and all qualities of nature, which are essentially imperfections. He is beyond all obscuration of ignorance. He is the soul of all and by him was spread out whatever is contained in the universe. He is of the nature of all auspicious attributes. By a tiny fragment of his power the aggregates of beings are lifted up. He assumes by his own desire a multiplicity of bodies. Thus he accomplishes the good of the whole world. He is a mass of only excellences like radiance, power, rulership, omniscience, vigour and strength. He is greater than the greatest. Not one of the imperfections like affliction ever approach him. He is the sovereign of the high and low. He is the Lord. He has the sum total of all subtle entities as his form. The gross universe is also his form. He is of the unmanifested form and manifested form. He is the Lord of all. He sees all. He knows all. He is all-powerful. He is the supreme God. That by which this pure, taintless, transcendent, faultless and undivided and sole reality is cognised, perceived and realized, is the only knowledge. All other processes of cogitation are mere ignorance (Vi. Pu. 6:5: 83-87)’. Thus the (Vishnu- Purana) exhibits itself as meant solely for the declaration of the determinate identity of the supreme Brahman.
All other Puranas with other aims must be interpreted consistently with this Purana. That they have other aims is discerned in the manner of their commencement. Whatever is found in them, altogether inconsistent with this Purana must be set aside as being of the nature of tamas.
Gaudiya Vaishnavas who are the worshippers of Radha-Krishna, on the other hand, consider the Srimad Bhagavatam as the best of all Puranas due to its central focus on the Lord Krishna's glories. This is what Chaitanya Mahaprabhu says about Srimad Bhagavatam in the Chaitanya Charitamrita:
sādhu-saṅga, nāma-kīrtana, bhāgavata-śravaṇa mathurā-vāsa, śrī-mūrtira śraddhāya sevana
One should associate with devotees, chant the holy name of the Lord, hear Śrīmad-Bhāgavatam, reside at Mathurā and worship the Deity with faith and veneration. (Chaitanya Charitamrita, Madhya-Lila, Chapter 22, Verse 128)
Vallabha Vaishnavas, who just like Gaudiya Vaishnavas, are primarily worshippers of Krishna. So it is pretty obvious they too consider the Srimad Bhagavata Purana the most important of all Puranas. In fact, Sripad Vallabhacharya himself wrote a commentary on the Srimad Bhagavatam, "Sri Subhodini". You can read it in Hindi here.
The Madhva Vaishnavas too, prefer the Srimad Bhagavatam to the Vishnu Purana, even though their Sampradaya is Vishnu-centric rather than Krishna-centric. In fact, Sri Madhvacharya himself wrote a commentary on only one Purana i.e. the Srimad Bhagavatam, "Bhagavata Tatparya Nirnayam". You can read it here.
The Sri Sampradaya believes Sri Narayana or Vishnu is Supreme.So they accept the Prasthanatrayi and Vishnupurana.
The Pushtimarga believe Sri Krishna is supreme and so they accept the Prasthanatrayi and Srimad-Bhagavatam ( As in Bhagavatam, Sri Krishna is the Supreme God).Pustimarga focuses on the leelaas of Sri Krishna. Soordas and other members of the Astachhapa wrote so many verses on Krishnaleela.
The Gaudiya Vaishnavas give equal importance to Radha and Krishna.But Radha is not mentioned anywhere in Bhagavatam explicitly.So they accept the Prasthanatrayi, Srimadbhagavatam and Brahmasamhita The last book Mahaprabhu collected from South India.They also give equal importance to Sri Chaitanyadeva as they accept Him as Sri Krishna Himself who was born to taste the Radha-prema and preach Nama-samkirtana.So they give equal importance to Sri Chaitanya-Charitamrita and Sri Chaitanya Bhagavata.
Sri Sampradaya worship Vishnu with Lakshmi, Madhwas worship only Krishna, preferably the balaka form that appeared to Sri Madhwa.Gauriyas worship Radha-Krishna and Sri Chaitanya side by side.Some sects of Gaudiyas also worship Nityanandadeva with equal importance.
The worshipped deities reveal the books they regard the most.
For the Bramha Sampradaya Vaishnavas (Madhva Sampradaya and Gaudiya Sampradaya) the Srimad Bhagvatam is the foremost. Sripad Madhvacharya wrote Commentry on only 1 Puran and that is Srimad Bhagvatam (The Bhagvat Tatparya Nirnaya) and In Gaudiyas Srila Jiva Goswami wrote the Sat Sandarbhas, Srila Baladev Vidyabhushan wrote a Commentry and Srila Prabhupad also.
The Rudra Sampradaya Vaishnavas also known as Pushtimarg or Vallabhacharya Sampradaya has Srimad Bhagvatam as the foremost and Sripad Vallabhacharya wrote a Commentry on Srimad Bhagvatam that is the Subodhini tica.
In the Sri Vaishnavas both the Vishnu Puran and Srimad Bhagvatam holds Great Position. There are 2 Commentries on Srimad Bhagvatam in the Sri Vaishnava Sampradaya.
I m not properly aware of the Nimbarki sect properly.
Overall The Srimad Bhagvatam is considered the Foremost and the spotless Puran. However other Satvik Purans like Vishnu Puran too hold great Reverence.
In Padma Purana Uttar Khanda chapter 236
मात्सयं कौर्मं तथा लैङ्गम् शैवं स्कान्दं तथैव च ।
आग्नेयं च षडेतानि तामसानि निबोधत ।।
Matsya Purana, Kurma Purana, Linga Purana, Shiva Purana, Skanda Purana, Agni Purana these six are Tamasic (236.20)
सात्विका मोक्षदाः प्रोक्ताः राजसा सर्वदा अशुभाः ।
तथैव तामसा देवि निरयप्राप्तिहेतवः ।।
Among these puranas Satvika Puranas lead one to liberation, Rajasa do not cause good > and tamasa puranas lead one to hell. (236.21)
Garuda Purana, Brahmakhand Chapter 1 —
तथैव वै वायुपुराणमास्त्रीव चाहुः सात्त्विकानीति लोके ॥ तत्रापि किंचिद्वेदितव्यं भवेच्च पुराणष्ट्के सत्त्वरूपे मुनींद्राः॥५१॥
Next in order of merit is the Matsya Purāṇa, then comes Kūrma Purāṇa, then Väyu. These three are also Sattvika Purāṇas. O best of sages, in the sixfold series of Sattvika Purāṇas, there is lot of knowledge to be discovered.
सत्त्वाधमे मात्स्यकौर्मे तथाहुर्वायु चाहुः सात्त्विकं मध्यमं च ॥ विष्णोः पुराणं भागवतं पुराणं सत्त्वोत्तमं गारुडं चाहुरार्याः ॥५२॥
Among the Sattvika Purāṇas, Matsya and Kūrma are inferior, Väyu is middling. Vișnu and Bhagavata are superior.