Password restrictions

Does anyone know why websites have password character restrictions? I don't mean minimum restrictions but maximum restrictions such as maximum length and disallowed characters.
Every time I come up with a password formula I run into a restriction that breaks it. The latest is that more and more websites are rejecting passwords with special characters like +=`#. Also finding more sites with a 12 character limit.

It is extremely difficult to come up with unique passwords that can be remembered when you are limited to 8-12 non-special characters.
YES I know about password vault programs and NO I don't care to use them! I like using the password vault in my head.

 

Ease of use and limitation of the software that is running on the sites. It is deemed sufficient to have up to 12 chars