Since 당신의 is optional in both sentences (당신 is rarely used in conversation though), the difference is in the choice of the particle, either 은/는 (topic particle) or 이/가 (subject particle). 은/는 is used when you talk about something because you want to express what you know or feel. It is a more organized and orderly way of saying something. For example, if you want to introduce yourself to someone, you'd say 나는 [Name]입니다, 나는 대학생입니다, 나는 스물두 살입니다, etc. 이/가 on the other hand is typically (but not always) used when you respond to something external, like describing what happens around you or answering someone's question. For example, if you saw an airplane flying, you would say 비행기가 날아간다 rather than 비행기는 날아간다, because you're not giving your own thoughts about airplanes but just describing a random occurrence. The above distinction is not always very sharp, however, so you can't just go by a simple rule. Sometimes either one might work equally well, and at other times using neither (i.e. leavie out the particle) might be more natural. This is a difficult subject. For "What is your name?", both 이름이 뭐예요? and 이름은 뭐예요? can be said, but 이름이 뭐예요? is more common. 이름이 뭐예요? sounds like you just saw someone and want to know their name, i.e. you're kind of responding to a situation casually. 이름은 뭐예요? sounds like a question you might ask when you're further along in the interaction, like you already know some things about the person and now want to know the name. It's like saying "(I see, that's great, but now) what is your name?". 은/는 has this nuance of "what about?" or "as for (something)".

What's your name? Vs. What's the name? The latter sentence is also okay.