Would a backpack or cargo rack be better for commuting? Right now I use a backpack to hold all of my supplies including spare tubes, pump, patch kits, tools, my change of clothes for work, etc.

This is the backpack I use:



It's about as comfortable as a backpack can get without getting into the hiking backpacks with frames. Altogether I would say it weighs no more than 10 lbs max. The muscles in my back get slightly irritated during my commute.

Would it be more comfortable to use a rear cargo rack on my bike instead? Will adding more weight to the bike change how the bike rides and feels?

EDIT:
After doing some research I came across this article that explains the different methods of carrying your gear on commutes and how it affects your body. It's worth a look.

EDIT #2:
Having tried both for a couple of weeks now, I can say that (for me personally) the backpack is the better way to go given the relatively light weight I have to carry.

If you do use a backpack, make sure that the straps are tightened enough so that it's positioned higher up on your back and to the point to where there is little to no movement on your back. This makes a world of difference when it comes to comfort.
I avoid backpacks where possible

1) Sweat - I get damp enough wearing just a cycling top.  Putting anything else on top makes it much worse.

2) Crashing - Occasionally I have carried a toolbox in a tramping pack.  One of the worries is "what happens if I end up rolling on this?"

3) Visibility - can be off-set with reflective tape or an overbag/wrapper/cover

You should at least try a rack or basket or frame bag.  See if you can borrow one for a month or so.