It's obvious when you think about it, but the biggest difference between an organ and a piano is the way their sounds decay.
A piano is a hammer hitting a string. The loudest sound is right at the beginning, and from there on the sound decays organically as the string returns to rest. If you let the dampers do their thing, the decay is shortened, but it's still there. The shape of the volume envelope of a piano is a sharp peak, followed by a progressively more gentle slope eventually settling back to zero.
An organ is the opening and then closing of a valve that pumps air into a pipe. There is no decay at all until you release the key, then the air and sound shut off virtually immediately. The shape of the volume envelope is like a table: straight up, then flat across the highest level, then straight back down to zero. If you use the swell pedal to open or close the enclosure around the pipes, you are affecting volume (that is, making the table top uneven), but you are not changing the fact that there is no gradually flattening decay. Of course, the reverberation of the room/church does it's thing after you release the note, but this is not at all similar to the way a piano string decays, and in any case you can't control it by, say, hitting the key harder.
This all means that you have to be very precise about something when you play an organ that you do not have to nearly as precise about when playing a piano: the moment you release the key.
On an organ: Release is an important as attack. Release is as much a rhythmical event as attack. And release changes the harmonic content suddenly.
Don't be worried about this. See it as a very cool additional feature of the instrument. You get two rhythmical elements for the price of one. It's really interesting to focus on the end of the sound as much as the beginning. It will also improve your technique and musicality on the piano.