The main difference is that AdAware is free forever, but doesn't provide any active real-time protection in the free version. AVG Antispyware offers real-time protection free for 30 days. After the trial period, it will only scan for and remove spyware. You must purchase a registration key to continue the real-time protection. They are both good at detecting spyware, but AdAware seems to find more stuff. Now, alot of this 'stuff' isn't necessarily bad stuff; many safe cookies are picked up by AdAware. Different spyware scanners will find and remove different things, so one isn't necessarily 'better' than another, they're just different from each other. When I run thorough scans, I use several different tools, not just one or two. As each different tool finds different items, the more different scans that are run, the more thorough the cleaning.... and when it comes to spyware scanning, I strongly recommend SUPER Antispyware. It's been my experience that it has one of the best detection and removal rates among the free scanners. But, no single application gets everything. Here's the link to SUPER Antispyware if you're interested: http://www.majorgeeks.com/SUPERAntiSpyware_d5116.html
They have a paid 'Pro' version that offers full-time active protection.