Old Cannondale MTB. 2nd set of wheels. Orig wheel has 8 speed cassette...2nd set has a 7 speed. Need advice
You do not want to run a 7 speed cassette with an 8 speed drivetrain. The sprocket spacing is not the same (5mm for 7, 4.8 for 8) so your indexing will be off, and you will run the risk of changing one gear too far and dropping the chain (no matter how careful you are, you will do this eventually). 

If the Schwinn wheel is a freewheel type, you can't put an 8 speed cassette on it. 

If the Schwinn wheel is a freehub type, you still have a problem. 7 speed freehub bodies (that the cassette fits onto) are narrower than those for 8 speed and above, and a 8 speed cassette will not fit (although apparently this can be hacked, see here https://www.sheldonbrown.com/k7.html#897).

Something else you want to do is check your frame's rear hub spacing, it's probably 135mm but the schwinn wheels may be 130.

If I were you I'd sell the Schwinn wheels and get a set of 8 speed compatible wheels to avoid the hassle of trying to make them fit.