This is a conversation from the Balarama thread that is too huge for the thread to not be derailed, so i am putting it here. I did not want to make it a private conversation, as i do think other Vaisnavas who seek this question to be answered can benefit from this information.
And so, what is this significant difference?Āveśa refers to the (usually) temporary appearance of a special form of nārāyaṇa (or other deities) in particular jīva. Aṁśa can be understood differently based on the context (even within specific context of avatāra), in general terms refers to a portion of which there can be different extents - partial, limited, absolute/complete etc. For e.g.., 1) paruśurāma, rāma, and pṛthu were vaiṣṇavāmśa-saṁbhūta, yet pṛthu was āveśa-avatāra whereas the former two were pūrṇāvatāras; similarly, 2) dūrvāsa, śuka, and aśvatthāma are rudrāṁśa-saṁbhūta - avatāras of śiva. All other deities too have avatāras, excluding brahma. However, there is a significant difference between 1) and 2) types of avatāras (perhaps different thread). But basically, all appearances of deities in forms other than their mūlarūpa is considered aṁśāvatāra.