You are correct that both models work on the same data without any problem. Both items operate on a matrix of user-item ratings.

In the user-based approach the algorithm produces a rating for an item `i` by a user `u` by combining the ratings of other users `u'` that are similar to `u`. Similar here means that the two users have a high Pearson correlation or cosine similarity or something of the like.

In the item-based approach we produce a rating for `i` by `u` by looking at the set of items `i'` that are similar to `i` (in the same sense as above) that `u` has rated and then combines the ratings by `u` of `i'` into a predicted rating  by `u` for `i`.

The item-based approach was invented at Amazon (http://dl.acm.org/citation.cfm?id=642471) to address their scale challenges with user-based filtering. The number of things they sell is much less and much less dynamic than the number of users so the item-item similarities can be computed offline and accessed when needed.