I would agree that the most common thing the snail symbols is slowness. I have also seen where it has been used to mean resurrection because of it's hibernating nature. I have also seen the snail used to symbolize persistence.
To most Americans, at least, the snail is used only as a symbol of slowness, like a turtle. We often use the phrase "at a snail's pace" to mean that something is going very slowly. I do not know of any other use of snails as a symbol that everyone in society would understand.
When someone mentions a snail to me, the first thing I truly think about is being 'slow'. Like running slow, doing things slowly, getting the job done slowly, but then I also think of the French food for 'escargo'. I guess it can usually mean 'slow and steady wins the race' as well. I also think of patience.
If you think about it, the snail can represent quite a few things, of course slowness being the first one that comes to mind. Secondly snails can symbolise a certain type of person, most commonly a cautious, slightly irritating person.