Well when you are playing an installed game the Xbox runs cooler, the Disk doesn't thrash madly and the gameplay is a lot smoother. Oblivion before and after installing has VERY obvious differences as does Fable 2.
+1
The biggest improvement (in my eyes) for installing to the HDD is cooling down the console. Just watching a DVD will produce heat. Playing a game does more. The HDD eliminates one huge source of heat generation in the console.
Frankly, if the game in question is old enough, maybe you can go to Half.com or eBay and pick up a cheap 2nd copy.