What is the difference between "dry run" and "rehearsal"?

You could say that a dry run is a type of rehearsal. Specifically, the type of rehearsal where you run through a show without stopping to actually work on issues, for the sake of testing where the ensemble's cohesiveness and preparedness is.

Rehearsal is a time set aside to rehearse, or, to work on a specific set of pieces, a show, or some other event. Sometimes, a rehearsal is involved in the sense of working on issues with the group, and addressing questions and concerns. On the opposite is a very hands-off type of rehearsal, which is the dry run. This is the rehearsal that can happen at specific determined checkpoints, to make sure that things are ready, or to assess all the issues at once before settling on a plan to address them in full before performance time.

The only difference that I can think of is that a dry run may not have a pre-defined outcome, whereas a rehearsal (of a play or a wedding ceremony, say) usually does have pre-defined course and outcome. A dry run of a military action may well have an unexpected outcome (but it is still a dry run).

As you say, the dictionary definitions are very much the same.