I've never opened a 3 box Cup case, so can't answer your question properly. I can say that the 6 box cases come as 6 boxes all in one case, so not two joined together 3 box cases, which "TheBostonKid" implies if I'm reading his answer correcly.
As for case hits, the 6 box cases usually had one really nice card (less often two...or less often none...speaking from experience...sigh) but I've heard of 3 box cases yielding a really nice "case hit" card as well (for example, an Ovechkin RC #/99 from an 05/06 Cup 3 box case). Keep in mind that they hand pack the Cup so sometimes defining a "case hit" may differ amongst some people. In general, I think the UD people packing out the Cup have improved with what constitutes a case hit, over the years. So in other words, the more recent years of Cup seem to be more consistent with each 6 box case having a true "case hit". I know no system is perfect so some cases still seem to lack any great card, but this seems to happen less in the last few years.
I'll ask that others comment from experience about the 3 box cases. I'm in Canada and we only had access to the 6 box cases from our dealers. I'll also defer to others who cracked the Cup in far greater quantities than I...you know who you are!
Whatever you do, buy a sealed case and from a reputable seller, so you avoid getting a case that's been stripped of its "case hit box".
