Actually it depends on the context. Don't you want to walk with me today? This would normally be when someone is walking maybe from one place to another. Don't you want to take a walk with me today? This would be when the main purpose is the walk, maybe for excerise or fun. Talking a walk rather than just the action of walking.

Tough question because, at first glance, the sentences seem seem synonymous. However, I see 3 different aspects to consider here. 1. Are they on a walk already and one walks ahead of the other sort of ignoring him/her? 2. If not out on a walk, is it a normal or pre-planned routine and the speaker is asking about a change in the routine? 3. If not out on a walk already AND walking together is not a normal or previously planned thing, then the second question might be an inquiry about the mood of the other person rather than being about the walk itself When spoken, differing emphasis in either sentence could clarify what the speaker is actually asking about. With no more information than the 2 questions written here, it's hard to say one is definitely worded better than the other.

They are both correct. But you're safer if you use the 2nd one.

I agree with Holly: "take a walk with me" is more commonly used.

both are correct, but the second one would be used more