"ludicrous" is a lot stronger than "ridiculous." Also "ridiculous" is much more common than "ludicrous."
In addition, "ludicrous" is really only used to say that something is almost impossible to believe. "A ludicrous claim," for example. But ridiculous has many other uses, e.g.:
1. She looks ridiculous in that hat. ( = She looks bad in that hat. You can't use ludicrous here.)
2. Ronaldo scored a ridiculous goal in that game. What a ridiculous play! (= What a great / amazing goal. This is actually a compliment! And you can't use ludicrous here.)
Hope that helps!