What you use is completely up to you.
You just need to realize that if you use an older technology there is less innovation going on with it and if less people are using it then there are less people that can help you out.
My last employer is still actively developing Classic ASP sites along with MVC versions 3 & 5. While they host all of these internally; any of these sites can go to just about any of the commercial hosting platforms supporting Windows.
ASP.NET Web-Forms require an MS stack to run on. It is based on making HTTP and the web as event-driven and having state on a stateless platform. While it was great in concept and could have been the wave of the future; the internet evolved in a way that didn't require state and events were driven in a way that did not require what MS offered.
MVC is a design principal with a firm Separation of Concerns. It is language and platform agnostic; which is huge as a project written for one platform can be ported to another without needing proprietary controls figured out.
MS decided to jump on the this along with joining the open-source community, so that developers would have options to work with. They have created a new track that doesn't require an all MS infrastructure and can have universal deployment.
Yes, there are pros and cons to both. If you want those you can read through this article which is still relevant today:
CodeProject: WebForms vs. MVC[
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