The basic difference is that weak references are supposed to be claimed on each run of the GC (keep memory footprint low) while soft references ought to be kept in memory until the GC actually requires memory (they try to expand lifetime but may fail anytime, which is useful for e.g. caches especially of rather expensive objects).
To my knowledge, there is no clear statement as to how weak references influence the lifetime of an object in .NET. If they are true weak refs they should not influence it at all, but that would also render them pretty useless for their, I believe, main purpose of caching (am I wrong there?). On the other hand, if they act like soft refs, their name is a little misleading.
Personally, I imagine them to behave like soft references, but that is just an impression and not founded.
Implementation details apply, of course. I'm asking about the mentality associated with .NET's weak references - are they able to expand lifetime, or do they behave like true weak refs?
(Despite a number of related questions I could not find an answer to this specific issue yet.)