DKB, we stayed at Tunnel Mountain II right in Banff. If you require power and water and you do not require a nice spot then Tunnel Mountain II is fine. Tunnel Mountain I does not have hookups but is a very nice campground with site in the trees and a little more privacy. TMI also has firepits, however you will need a daily permit to have a fire and buy the wood on site.
Just outside Banff there is a great little campground beside Two Jack lake which is part of Lake Minnewanka. No services but the spot is fantastic.
You can reserve at Tunnel Mtn but I don't know about Two Jack. All the information can be researched here:
or try toll free at 1.877.737.3783
Have a great time on your trip and I hope you enjoy Banff. It is a beautiful place but be prepared for huge crowds if you will be there on a weekend.
Regarding Jasper: We have stayed at Whistlers campground before and it's great. This year we will be trying out Wapiti campground. You can also research Jasper campgrounds at the above link.