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 user's ratings have a high Pearson correlation or cosine similarity or something similar.
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 except now we'd be looking at the ratings that items have received from users) 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 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.
