Rough is usually more than just tired. It is more feeling a bit bad. It's commonly used to refer to being hung over, but could also mean you slept really uncomfortably, or perhaps you haven't had a shower for a few days.

It's not a mainstream usage but it means "feeling unwell" or "feeling down". By extension, you could take it to mean "feeling tired", I suppose.

Gary has a good answer. In my opinion (American usage), it is used to describe "feeling terrible" after excessive drinking. I don't think I've ever heard it used in another context. The Oxford Learner's Dictionaries are an excellent, online resource. They give a British usage as "not feeling well." https://www.oxfordlearnersdictionaries.com/definition/english/rough_1?q=rough not well ​(British English) not feeling well You look rough—are you OK? I had a rough night (= I didn't sleep well). There is also the related British phrase "to sleep rough" which means to sleep in the streets.