Was the shower screen screw (and screen itself) difficult to remove? That might give a clue as to what might be going on. How old is the machine and what is the maintenance history? If it hasn't been cleaned well over its life -- and especially if the shower head screen and plate haven't been removed and cleaned in a long time -- there's a good chance a lot of crud has built up in there. And by crud I mean old coffee grounds/oils, scale from water minerals, and aluminum oxide.
If I was sure that all screws were removed, I'd probably tap gently around the perimeter of the shower plate with a ball peen hammer, or maybe a blunt-edged chisel if there's not enough room to work with the hammer.
