The primary difference is the adjustability and the quality that goes into the drive mechanism. Spinning bikes, as the name implies, spin a weighted flywheel using the power of your legs. The tension you set, in conjunction with the weight of the flywheel, determine the resistance you feel. This provides a much more realistic effect in terms of how it uses your muscles. A old school excercise bike isnt as good at effectively targeting the muscle used and doesnt provide as much of a realistic cycling feel.
That said, if your goal is just to get a workout in you arent going to miss much. A competitive cyclist would notice the difference but a casual rider just looking to burn some calories probably will notice only the terribly uncomfortable saddle of an old excercise bike.
Think of it as the diffference between running wind sprints vs. running parachute assisted acceleration sprints.... Unless you're a pro athlete working on dropping your 40yd times, wind sprints is just fine. Similar situation here. If you're gonna be heading to the races, start spining.. If you just want a workout save the money.
PS- If you get serious about riding and you really want the best set-up, IMO its using your road bike on resistance trainer like the cycleops fluid2 or something similar. It lets you keep your cockpit setup (duh, its your bike) but gives you the indoor training option for those shoddy days when you dont want to go outside and ride.