The big thing with Inn is that it can shave a number of turns off where your engine might hit, but is not yet assured. I get the most mileage out dumping engine components once or twice into the top deck (even at the expense of getting a better $5 village like Bazaar). As such it tends to be a bit better when I already have a lead and I want to tamp down variance. Things like shuffle up Inn/Village/Smithy/Steward are quite powerful at cycling the deck and getting stuff trashed faster.
Inn is also better than a second Curser or Looter in the mid-game. Winning the Curse split 6-4 is a pretty big thing with limited/no trashing and absolutely huge if that is coupled with no +buy/+gain. A 2 VP swing when you can't gain more than one VP card a turn is effectively a duchy that cycles (which is not bad at all in mucked up decks like those circumstances). The question of do I get a second Witch or Inn the first one up twice is a bit difficult. If you can manage an engine, Inn is normally better, but if you can't somewhere around Curse 4 it tends to be better to go for the the Inn. Definitely if you only will use the curser once more, Inn is better and depending on setup, it might be better even if you expect to curse just twice more. Rarely is Inn a good call if you expect to curse 3 more times.
As a village itself, it gets better with stuff that likes cycling and makes draw have a low opportunity cost. Shuffing Lib/Inn to the top lets you draw a lot of good cards and have very high odds of pulling cards currently in play (like say an Expand) next turn as you Lib/Inn will let you have a search space of 11(9). Inn drop in hand size is actually beneficial when you really want its on gain ability (early engine setup where you need to cycle to power cards) and draw is cheap. If you can just spam simple draw (like Smithy) cheap, Inn isn't bad either. 2 Inns/2 Smithies will draw draw 10 and leave you with 7 cards; you can do quite well at quickly setting up an engine with Workshop/Inn/Smithy and quickly draw all cards.
Inn, like all cards with on-gain bonuses, gets better with scaling TfB. It makes excellent Remodel/Bishop/Apprentice fodder as it is a card costs $5, but doesn't play at $5 value (Inn without on-gain should definitely be sub-$5).
Inn has a rare niche in allowing you to set up an "engine" or combo deck in the midst of otherwise impossible bloat. Like all attempts at this (Wt, Scheme, etc.) this is normally weak. But something like University/Rabble/Inn can allow you to use a $5 to place half your Unis/Rabble into an empty top deck, play them and then gain an Inn to top deck the other half. Yes this means you need 2x as many components ... but it can allow you to go for some engine combo after your deck has been bloated (e.g. Ambassador in 4er). Inn is also good at enabling Countinghouse engines. Inn + Chouse + Cellar + 2 other actions means that you can ensure every turn begins with a bunch of copper bloat in the discard, you can draw 15 cards with Chouse, and then draw your real cards without too much trouble. Inn also helps in that its discard ability allows you to seed your discard with the cards you need for your next turn for this sort of thing. Yes it costs $5 (via a buy or a gain) to do this, but that isn't that bad if you can't make a strong "engine" any other way.