In one sense, there was not much difference between these two kinds of weapons. Both were used for launching rockets. Rockets were not terribly accurate weapons, but they were able to carry relatively large payloads and were fairly portable. So, in this sense, these were very similar weapons. This similarity even extends to the fact that both were given nicknames based on the screaming sounds the rockets made.
There was one major difference in the appearance of the two weapons and in the exact way they functioned. The German weapons were adapted from mortars. Therefore, the rocket came out of a mortar-style tube when it was fired. By contrast, the Katyushas were launched off of rails. This was not a major difference, though, in terms of the actual utility and effectiveness of the weapons.