How can one equivalent statement be stronger than another?

It’s been a long time since any mathematical content was posted on this blog. This is in part because I have been diverting my mathematical efforts more to the Tricki (not to mention my own research), and indeed the existence of the Tricki means that this blog will probably become less active, though I may publish some Tricki articles here as well. But there are certain quasi-philosophical questions that I want to discuss and that are better discussed here. I have already written about one of my favourites: when are two proofs essentially the same? Another that is closely related is the question of how it can be that two mathematical statements are equivalent to each other and yet one is clearly “stronger”. This phenomenon will be familiar to all mathematicians, but here is an example that illustrates it particularly well. (The example is also familiar as a good example of the phenomenon).