Cats can catch small birds while Sims can befriend big birds (and teach them to talk). Sims can catch small ones too.
My apologize for ugly photo editing
Cardinal - uncommon, 123$
Bluebird - rare, 658$
Pigeon - common, 20$
Parakeet - common, 8$
Goldfinch - uncommon, ? ? ?
Nuthatch - uncommon, 217$
Lilac Breasted Roller - ? ? ?
Catalina Macaw (Big bird) - uncommon, 138$
Spotted Sixam(Big Bird) - rare, 1053$
Big birds were caught by my Sim, and she teach Spotted Sixam to talk, so that might increased his value. All same type critters my cat caught were the same value (all Cardinals were worth 123$ etc.)
Once I got them all, I'll try to put together more neat post with all of the info
The Sixam is probably the hardest bird to get but I bet it's still doable.
That was the first bird my Sim saw and first wild animal she befriended, so it is not really hard (only other friend she has is her hunter cat which I created in CAS). I never saw a falcon, crow or any other bird other than those in pictures. I'm still looking for a way to locate big birds since cat can't catch them and they can't locate them neither. So far I find them just by accident, while looking for other things around.
Here is Spotted Sixam playing with my Sim: