Zack's spare uniform doesn't fit Cloud perfectly; it's a little baggy on Cloud's smaller frame.
Anyway...
Zack was infused with Jenova cells, only after joining SOLDIER. Iy's a normal part of the process used to make SOLDIER members so strong. However, people have to be pretty strong already before they can join SOLDIER, so they can resist the 'brainwashing' effects of Jenova's cells.
Sephiroth was unique, he was the 'ultimate' blend of Jenova's cells and human life. Shinra wanted to re-create this success, to "clone" Sephiroth, so to speak. To do this, they exposed regular people to Mako energy and Jenova cells - two key steps in making SOLDIER members, but with more serious effects for regular people. Most became those black-cloaked "zombies" when Jenova broke free of Shinra's imprisonment; Sephiroth began to guide them to the North Cave and use them to do his dirty work. Cloud was busy living out his ex-SOLDIER 'daydream', so the effects on him were different.