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Difference between:

Are you coming with us? And Are you going with us?

Don't they essentially mean the same thing?

Thank you.
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If the friend that you're talking to is at home use come. Come indicates movement in the direction of the speaker/listener.
Comments  
"Are you coming with us?" is an invitation or encouragement to personally accompany someone. "Are you going with us?" feels more impersonal -- like you will be part of a larger and more anonymous group that just happens to be going in the same direction.

If you have already started the journey (for example, you've walked ahead and you are encouraging someone to catch you up) then you must use "coming".
Students: We have free audio pronunciation exercises.
Hello,

I'm on the phone w/ my friend who's at home. Could I say:

Could I come over to your place to take a shower after I run the marathon. Or should it be:

Could I go over to your place to take a shower after I run the marathon.

Are they both correct?
 Ivanhr's reply was promoted to an answer.