to compress is to apply pressure to push something down but compression is only temporary once you remove the pressure it goes back to its former state. Compression is used to stop bleeding, by pressing firmly over the injury. If you compress material like sponge, it goes down into a smaller space, but will spring back again when released from compression. To shrink something is to make it smaller permanently by altering its state.

Compress = 縮壓 Shrink = 縮小 所以分別在"壓"的部分. The difference is idea of "pressure" in "compress", where you see the same "press" word in both words. Shrink does not imply any sort of external force.