There are three main differences: URL mapping, separation of logic from presentation, and strong typing.
URL Mapping
With classic ASP there is a smooth transition from writing HTML pages to writing HTML pages with dynamic content. As with static HTML files, each URL has a direct mapping to a file in the filesystem. The same thing is more or less true of ASP.NET, for what it's worth.
In ASP.NET MVC, each "family" of URLs maps to a Controller object (stored in the /Controllers directory, by default), where each member of the family calls a method when accessed. At the end of each method (typically), you tell it to render a particular view (stored in a folder named after the controller in the /Views directory), which is a lot like a classic ASP page with all of the logic separated out.
This gives you logical and SEO-friendly URLs and groups related functionality together.
Separation of Logic from Presentation
In classic ASP it's common to find pages where a bit of HTML is included at the top, and then a database connection is opened and some things are read from the database while being output to the user, and then some more html, and then another database statement, and so on.
In ASP.NET MVC, your business logic (e.g. validation) goes in the model layer (you can choose from one of several dozen, but popular choices are LINQ-to-SQL and LINQ-to-Entity-Framework), your human interface logic goes in the controller (e.g. populating a State/Province menu based on the Country selection), and your presentation (the actual HTML you can hand to a designer to edit) goes in the view.
Aside from keeping things organized, this helps a great deal with being able to write automated tests for things. You can send a mocked-up object to your view and make sure it looks good, you can send bad data to your model and make sure it complains, and you can make sure that the object your controller sends out to your view is consistent with what it reads from the model.
Strong Typing and Compilation
ASP.NET is strongly typed and compiled. This is a double-edged sword. On the one hand, it will catch a lot of stupid programmer mistakes at compile time. On the other hand it, that means you're left with "infinity minus one" possible errors in your code (unit testing can make it infinity minus some larger number). Also, you'll have to do things like:
if (MyArray.Length > 0)
rather than
if (MyArray.Length)
But IMHO that's a small price to pay for the speed and sanity-checking you get from strong typing.
The bigger downside to compiled languages in a big framework is that deployment becomes much more of a production than it is with something like Classic ASP. You can't just copy a couple of files to the web server to update your app. You typically have to take the webserver down (hopefully you have a redundant pair) and recompile, which can take minutes.