What are the pros and cons of Bhakti Yoga and Gnana Yoga?

Firstly pros and cons are based on one’s svabhāva — one’s natural disposition. We all operate on three levels, physical (karma) emotional (bhakti) and intellectual (jñāna) — so the 3 yogas address these three aspects of our being.

Each and every individual operates with different combinations of these three factors; i.e. some people are more practical, some are more emotional and some more intellectual. But these 3 cannot be practiced in isolation so for example in order to accomplish any task you require an emotional commitment to that project (bhakti yoga). You then require to obtain some knowledge about the details and prospects of the project (jñāna yoga) and then you need to apply yourself (karma yoga). This is why any successful project will require people who are expert and dedicated to each of these three areas (adhikāra).

So the yoga that one practices is based on one’s svabhāva - not on one’s personal whim.

Rāmānuja ācārya taught that jñāna yoga practice is specifically the intellectual meditation on the formless and all pervading ātman within oneself. Whereas bhakti yoga is loving meditation upon the form of Lord Vishnu within one’s heart. Both are meditation techniques but one is formless and the other is with form - see 12th Chapter of the Gita.

Rāmanuja ācārya also taught that the basis of both these types of meditation is KARMA YOGA — being socially engaged in the world, performing one’s bounden duties as a valued and proactive member of society.

Bhakti Yoga and Gnana Yoga, Yoga here means method using for spiritual salvation. Bhakti means, total surrender to self ( supreme God ) Swa swarupa Anusandhana ithi Bhakti. What is true Bhakti, is one who has good understanding about supreme God, and he leads and surrender and offer everything to God. Bhakti Yoga has many levels, where it leads total surrender to Supreme God, Saranagati. Here Bhakti Yoga is little tricky often it can lead to a blind folded belief, this will not bring spiritual progress as it is not well guided with knowledge from a Guru.

Where as Gnana Yoga, Which can be followed by highely advanced spiritual seekers, and it is not just the intellectual path, which we can preceive from common intelleligence. Gnana Yogis, Intellect is always guided and connected supreme, so his understandings, mental logics will be fully different from a common man's intellectual layer.

So there's no pros and cons for Bhakti or Gnana, but these Yoga or methods can be practiced based on the spiritual maturity or eligibility of oneself.