I wanna build a custom server for some project and I don't know whats the difference between asyncore and asyncio server, what is better to use, and why
asyncore is the "old", Python 2 way of doing some basic event-driven servers and clients (in other words, for doing asynchronous IO).
asyncio is the new, Python 3 module that provides a whole framework for doing asynchronous IO in general. It has much more features, including support for coroutines, which allow you to use keywords like async def and await, which improve readability of asynchronous code.
In summary, asyncio is the way to go. asyncore is not recommended in Python 3, which new projects should be using instead of Python 2. If you are stuck with Python 2, asyncore is a sane choice.