Yes, definitely. There is just too much movement even in the very best trainers on a road surface.
Think about it - the very best most modern tracks have a lot of engineering going into them. Things like MONDO, new developments (I am out of date though). It is going to be designed to be the fastest surface - although different tracks may have different strengths, like one being very good for short sprints and so on. Spikes have also progressed over the past years, although the metal spikes themselves haven't changed vastly to my knowledge.
Spikes are bound more tightly to your foot, but it's the plate and the spikes themselves that IMO make the most difference. On the road there would be a small bit of slippage with each step but with spikes your step is locked in. They are designed to go fast.