Warning: "feature length article" aka rant ahead...
I don't particularly think of Sapp as a "nice guy", but I also don't think of him as the type to celebrate injuries (that would be the Eagles fans in the stadium at the last Bucs-Eagles game who cheered when they realized that it was Sapp on the ground injured when his eyes got gouged there). There was an interception on the play, and defenses tend to jump around like fools when that sort of thing happens. I'm sure Sapp was celebrating the fact they just made a play that might turn the game around, and probably not thinking of Clifton much (certainly not at first).
We don't really know what was running through Sapp's mind. Maybe he cared zero about whether he put Clifton in the hospital or not. Maybe he cared a great deal, but thought it would be rude to go out there and get in the doctor's way trying to "see if he was okay" (especially since the answer was clearly "no"). My guess is that he cared a little bit, but purposely put it behind him so it would not impact his play. I've seen teams carry a lot of guilt after inflicting such injuries and losing the game because they are no longer 100% into it (while the other team is 110% into it and wanting revenge).
I don't really know what was going through the coach's mind either. At best, he was mad at Sapp for having the temerity of trying to go over to shake his hand after hurting his player earlier and celebrating that fact. At worst, the whole thing was a calculated move to create some "Bucs hate" for the next game. I wouldn't put it past some coaches to do something like that (and I don't know enough about the Packers coach to judge that).
Now that the Bucs are finally good (I think), I'm starting to re-examine my bad feelings towards teams like St. Louis, Minnesota, Dallas, San Francisco, and yes Green Bay in years past. I guess opposing teams feel they need to use hatred to motivate their team to beat the current "800 pound gorilla". These days, that title's been bouncing around a lot, and today Tampa Bay holds that hot potato. The longer they hold it, the more they will be hated (2 eye gouges this year on Bucs players, and I have the impression it's pretty difficult to do them accidentally with modern football helmets).