My intuition says that exhibit has a more concrete, physical meaning than exhibition.
The -tion suffix itself is used for nominalizing verbs, for example:
- immigrate -> immigration
- mutate -> mutation
- act -> action
An exhibit would be a physical display of something - something you can actually visit and look at. Conversely, an exhibition is merely the action of displaying something in a more abstract sense.
So you could hypothetically have an "exhibit of talent", but it would be very peculiar. You could buy tickets for it. Maybe there would be jugglers and things of that nature. It might be advertised in the newspaper. An exhibit has a physical presence that an exhibition doesn't have. An exhibition might occur in a physical space, but it is not physical itself.
So I think the second sentence "It was the best exhibit of talent I've ever seen." is feasible and possible, but strange and why you won't find any usage of that phrase.
Another example of something similar would be the difference between a proposal and a proposition.