SharePoint is an application that sits on top of ASP.net. They do override a lot of ASP.net built-in functionality (they have their own .aspx Parser and Virtual Path Provider for example).
With ASP.net you have a very well documented, battle-hardened, mature and stable platform with a good API.
With SharePoint you gain a poorly documented, bug-ridden, very limited application that handles a lot of features that you would have to code yourself (e.g., User Profile Management, Document Organization and Versioning and Social Features like Commenting and Tagging), although for the most point SharePoint handles them really poorly and does not allow you to override them, which means that you spend a lot of time rewriting them anyway and trying to integrate them back.
Basically my advice as a SharePoint developer: Use it when you absolutely have to, avoid it whenever you can and stay with just ASP.net.
SharePoint is good as a simple document management and very light social system. You can quickly customize smaller parts of it and add a lot of value to your company. But in the moment you need something that even only slightly different from what Microsoft envisions, you hit a wall that you can't pass. It's great for what it does, nothing more, nothing less.
As a ASP.net developer, you can get an idea of SharePoint using the following links,
For working with Visual Studio to develop SharePoint solutions, refer the below URL's