The same problem pretty much plagued me when I was competing. At my best I could snatch 142.5kg with straps but in competition my best was 125kg. I have small hands and always had grip problems with the snatch. For that reason I was always feeling stronger with straps.
The mistake I did back then was to continue training with them every time I snatched. Most elite lifters do wear them in training when doing snatches, so I reasoned that it was alright to do so.
In retrospect this was idiotic: elite lifters have great grip strength. So they can use straps in training because they know their grip is strong enough (they never use straps on the clean BTW, so the grip still get worked). The reason they do not wear straps was not to pull more weight, but to avoid bruising the hands (which can happen when you training on the snatch 6 days a week, twice a day like these guys do).
Lifters with good grip strength CAN use straps on snatches and pulls in training (but not on cleans or deadlifts) and get away with it. Lifters with a weaker grip can't. They must work on solving the problem.
As Mike recommended learn the hook grip... and most of all LEARN TO GET COMFORTABLE WITH IT (it's quite painful at first). BUT I would recommend using the hook grip ONLY ON THE HEAVIEST SETS. Use a non-hook grip until you start to lose your grip, then switch to a hook grip (this is what Tommy Kono did).