This is a good question.
Rhythm, at its most basic, can be described as a movement with uniform recurrence of a beat or accent. Of course, there can be different rhythms and complex ones, but the key is the uniform recurrence. We have rhythm in poetry, songs, and other artistic outputs.
Rhyme, on the other hand, is a matching of sounds. Usually there is a matching of the ending of sounds, but there can be more complex rhyme patterns, such as when the matching of sounds takes place in the middle of the word. We can see this example, for instance, in Latin poetry.
An example of rhythm can be seen in the Homer's Iliad, where the poetry is written in dactylic hexameter. Rhyme can be seen in the silly sentence: "The cat sat on the mat."