File extensions for cryptographic certificates aren’t really as standardized as you’d expect. Windows by default treats double-clicking a .crt file as a request to import the certificate into the Windows Root Certificate store, but treats a .cer file as a request just to view the certificate. So, they’re different in the sense that Windows has some inherent different meaning for what happens when you double click each type of file.
But the way that Windows handles them when you double-click them is about the only difference between the two. Both extensions just represent that it contains a public certificate. You can rename a certificate file to use one extension in place of the other in any system or configuration file that I’ve seen. And on non-Windows platforms (and even on Windows), people aren’t particularly careful about which extension they use, and treat them both interchangeably, as there’s no difference between them as long as the contents of the file are correct.
Making things more confusing is that there are two standard ways of storing certificate data in a file: One is a “binary” X.509 encoding, and the other is a “text” base64 encoding that usually starts with “-----BEGIN CERTIFICATE-----“. These encode the same data but in different ways. Most systems accept both formats, but, if you need to, you can convert one to the other via openssl or other tools. The encoding within a certificate file is really independent of which extension somebody gave the file.