It means the same thing, but for a thorough translation you should use mi-aş, "mi" being the pronoun which refers to the subject. You can also say "Mi-aş dori să rămân cu tine", without the subject "Eu" (this form is much more common when speaking, and when you actually say the "Eu" as well, it usually means you emphasise it)
I think there is a slight difference, but it probably has to do with the question to this answer. For instance, if somebody asks you: "Dar tu ce ți-ai dori?" (meaning But what would you like? emphasising the "you") the correct answer is: "Eu mi-aș dori să rămân cu tine! ". If there is no question, but someone just puts out his/her feelings, then the most used form is: "Aș dori să rămân cu tine".