Originally Posted by SkyDog75

You may find the shifting to be satisfactory, but it's not a perfect match. Shimano 7-speed cogs are 5.0 mm apart, center to center. Shimano 8-speed cogs are spaced 4.8 mm apart. Each shift you make away from the cog where you aligned indexing, the derailleur will get 0.2 mm farther out of alignment. If indexing is perfect at one end of the cassette, it'll be off by 1.2 mm at the other end, which is definitely enough to cause poor shifting. If you adjust indexing using the center cog on the cassette, the chain will be 0.6 mm too far inboard at the biggest and smallest cogs, which isn't as bad, but could very well be enough to cause hesitant shifts or chatter. Then if you use the barrel adjuster to make that right, you cause the exact same problem at the other end of the gear range.

As for adjusting limit screws... Whenever you install a different wheel, even one with the same number of cogs, you need to adjust (or at least check) the high and low limit screws.