However, with the way things are currently going, are we using WebAssembly to reinvent the concept of Java applets? It all sounds very familiar. And if this is the case, will it run into similar problems?
There are, of course, a lot of similarities between the Java or .NET runtimes and WebAssembly, and other comments made excellent points about the lower level nature of Wasm, or the isolation sandbox.
I would also point out that there are significantly more languages that either have support for compiling to WebAssembly today, or that have started adding support (in the last month alone there were initial announcements for Python, Ruby, and .NET, for example).
Because it's lower level it's a more viable compile target for things that couldn't easily be compiled to Java, like C++ for example.