"would appreciate hearing your thoughts" phrase

I would appreciate hearing your thoughts on this ...

In what situations is this phrase usually used? Formal, informal...?

In contrast to:

I would appreciate hearing your opinion on this ...

They mean different things, as I am sure you realise, though they can be and are used in a similar sense.

'Opinion' is probably the more 'formal' of the two. Asking someones 'opinion' suggests you are looking for a more considered, more formal reply.

Asking for someone's 'thoughts', is more a case of getting their initial reaction, a sounding of their views.

I would appreciate hearing your thoughts on this ...

This goes for a person you like, or are interested in, or you care about.

And that's why you want them to open up and share, for whatever reason.

I would appreciate hearing your opinion on this ...

This goes for a person who is considered an expert in that field or at least an intellectually proven personality.