The short answer is "no, probably not". The longer answer is "you might, but it's probably mostly in your mind". Your Specialized is already a decent bike. There's no magic in going from what you've got to another bike with a full carbon frame. If you were coming from a stupidly heavy, rigid, or otherwise uncomfortable bike then you might notice a difference. But you're not, so you won't be losing vast amounts of weight (not enough to really notice, I'd suspect), or swapping a horribly uncomfortable frame for a vastly more compilant one.

If you want the carbon frame, then definitely go for it. Just don't expect a "night and day" difference between it, and what you've already got.