Connotation, mostly. The words only differ in the implied meaning, and the word choice would change to reflect the intended audience. To "comply with" is more positive and unoffensive. You are asked to comply with the guidelines as a customer in a store. To "abide by" is more negative and professional. You're obligated to follow guidelines whether you like it or not as an employee. Either way, failure to follow the guidelines will get you in trouble.

Basically the same meaning, though "comply with" implies to me something you have to do affirmatively, whereas "abide by" is a little more general (could be something tou have to do or something you must not do).