Mono and coreCLR are two alternative implementations for .Net runtime. Just like Firefox and Chrome are two implementations for various web formats, or CPython and PyPy for python language and many other examples.
Note in this comment I might be slightly imprecise with the use of naming since the specification, standard library, runtime can be separate projects each with its own name. But often multiple of those parts are made by same people so it can be a bit confusing especially with the way Microsoft names them with various capitalization.
Mono started as open source implementation of .Net and means to run C# on Linux when the Microsoft implementation of .Net was fully closed source and worked only on Windows. At that point it made sense for Unity to use whatever open source implementation of .Net they had access to which was mono. Years have passed and a lot of things have changed. At some point Microsoft bought Xamarin(the company that is main contributor/maintainer for mono project).
On the other side coreCLR is latest implementation of .Net from Microsoft. Unlike earlier versions of .Net from Microsoft it is open source and compatible with Linux and macOS.
As with many languages one of the implementations (the one made by inventor in this case Microsoft) is the leader, while others are playing catch up both in terms of new language feature support and also quality and tooling. Such situation is hard to avoid because developers of one implementation are making whatever they want while being sponsored by big company. While the others are trying to implement something that's more or less compatible with leader while having much smaller budget.
In the early days mono at least had the benefits of being open source and crossplatform compatible. Now they have lost this advantage. Even worse, the main contributor of mono is owned by Microsoft so there is high chance that they will slowly kill it in favor of their own implementation. Since the coreCLR is the primary implementation made by Microsoft it will more likely have much better integration and compatiblity with all the developer tooling made by Microsoft (Visual Studio, debugger) , better compatiblity with various third party .Net libraries, better support for new language features.