What is the difference between Mantra-siddhi that is described primarily in the Tantras as success of the sadhana and Bhagavat-prapti which is thought as the ultimate goal in the dvaita bhava upasana?
As per Tantra-Saara, signs of Mantra-siddhi, as mentioned in this answer : What are signs of mantra siddhi? are:
Fulfillment of desires without any real efforts is the principle sign of Mantra Siddhi. Other signs are- Defying death, getting visions of Gods, ability to enter other's bodies, ability to invisibly enter others' mansions, ability to move along the sky, ability to go wherever one wants, ability to meet with the goddesses that live in the sky and hear their voices. A Mantra Siddha person will be able to see holes on earth (Bhuchidrah darshana) and will get to know the principles of the earth. He will have great fame, will obtain vehicles, clothing and other such luxuries and will live an eternal life (exceptionally long?). He will be happy, and will be performing spellbinding acts everywhere. He can cure diseases simply by his vision and likewise can make poison ineffective. Without any real efforts he obtains four-fold mastery over the scriptures. He further obtains indifference (VairAgyam), a longing for liberation (Mumuksha), a nature that loves to sacrifice oneself for others (TyAgitA) and also the power of controlling all (SarvavashyatA). He obtains the practice of the eightfold Yogas, obtains freedom from desires of enjoying the objects of pleasures. He obtains compassion for all beings, becomes omniscient and also obtains similar other qualities. These aforementioned ones are all signs of a middling nature Mantra Siddhi. An inferior (Adhama) kind of Mantra Siddhi's signs are- Acquisition of fame, vehicles, clothing etc, long life, becoming dear to the King and his family; becoming dear to all people; obtaining the power of controlling others; attaining massive wealth, properties, wife, sons etc. These signs are manifested during the first phase of Mantra Siddhi. But one who has truly obtained Mantra Siddhi becomes just like Lord Shiva. There is no doubt in this.
So Mantra-siddhi generally is associated with bhukti and Krama-mukti. Fulfillment of desires is possible when there is at least some desire. Shiva also says that Tantrik sadhana leads to Krama-Mukti .
Shiva appears to them in form of Devata and gradually takes them to the desireless stage. (Reference :Satramukak Bharatiya Saktisadhana, Upendrakumar Das,RMIC, Vol. 1.
Bhagavad-Darshan on the other hand is defined in scriptures like Srimad-Bhatavatam. Sri Krishna says to Uddhava in Srimad-Bhagavatam(11/14/12)
mayi arpita-aatmaanah sabhya nirapekshasya sarvatah/mayi aatmana sukham yat tat kutah syaad vishayaatmanaam// (11-14-12).
meaning : Those who have surrendered to me fully and been freed from all worldly things, see Me appearing in their hearts.That bliss is never attained from any getting any worldly matter.
nirvinnaam jnana-yogo hi nyaasinaam iha karmasu/teshu anirvinna-chittanaam karmayogastu kaaminaam (11/20/7).
meaning : Those who are not yet detached from karma and have desires in mind should adopt karma-yoga.Those who are indifferent to the fruits of karma and reject [vaidhi] karma, should adopt the Jnana-marga.
yadrichchayaa matkathaadau jata-sradhdhastu yah puman/na nirvinno nasishakto bhaktiyogo'sya siddhidah//
meaning : Those who are are neither fully attached to nor detached from the fruits of karma would succeed by bhakti-yoga.
Sri Bhagavan therefore shows all three paths of jnana, karma and bhakti. Srimad-Bhagavatam also says that
vedyam vaastavam atra vastu ivadam tapatryonmulanam (1/1/2)
meaning : That has to be attained which is the Real Truth, Real Good and which removes three kinds of suffering.
That Real Truth is named by Srimad-Bhagavatam as Brahman, Paramaatman or Bhagavan. for Jnanis, yogis and devotees respectively.
The signs of attaining the Real Truth or Bhagavat-prapti is:
vidyate hridaya-granthis chhiddyate sava-samsayah/ kshiyante chaasya karmaani mayi driste akhilaatmani//(12-20-30).
All the ego goes away, all the doubts vanish and all karmas are destoyed if one attains Me as the Atma-svarupa.
So seeing God is equivalent to knowing of the Self. There is no desire to attain anything as nothing more has to be attained. The aspirant attains Jivan-mukti.