So far, there are two completely contradictory answers, one saying they're complete synonyms, and one saying 'uncommon' is 'more common' than 'rare'. One has received a downvote for being 'misleading'.
My instinct was that @Jaime was right, but on reflection I realised that as a negative form of 'common', uncommon might acquire slightly different meanings according to context. However, this would be the exception to the general rule. I can't think of a sentence off the top of my head where the difference is significant. You probably need to use both words in close proximity before any distinction would be relevant.
The crested newt has become uncommon in recent years, but is not yet considered rare.
But in both my example and @Biblasia's, 'rare' is being used in a more technical and scientific sense of 'rare animals', and it's the context of the usage that creates a distinction in meaning, not the words themselves.
So I actually think both contradictory answers can be correct, in some circumstances. However, I think as a general rule of thumb, you can treat 'uncommon' and 'rare' as exact synonyms, and therefore Jaime is more correct.
I plead guilty to not having checked the dictionary first, but, the 2022 edition of the American Heritage dictionary says this:
un·com·mon (ŭn-kŏmən) adj.
- Not common; rare.
- Wonderful; remarkable. un·common·ly adv. un·common·ness n.
rare 1 (râr) adj. rar·er, rar·est
- Infrequently occurring; uncommon: a rare event; a plant that is rare in this region.
- Excellent; extraordinary: a rare sense of honor.
- Thin in density; rarefied: rare air.
It seems to me that the dictionary considers Jaime's answer closer to the normally-understood truth than Biblasia's.