The do mean the same, but the former is just a very informal way of speaking.
If the person listening to you is a native English or long-time ESL speaker, then he or she would have little to no problem putting your reply into the proper context.
Q: Are you are home? A: I'm not at home
Q: Are you are home? A: I'm not home ["right now." - will be understood]
The preposition "at" is used to indicate presence or location, so both Andre and Ahmad above are correct in choosing "I'm not at home." It is the more grammatically correct or proper one.
However, their logic as to why one can't say "I'm not home" is also wrong.
No one who says "I'm not home" is equating himself (or herself) to a house.
#1: [calling #2] Phew! I'm really glad I caught you at home...
#2: What? I'm not home. This is my cell phone! [again, leaving out "at" would be understood and in this instance, "home" was used instead of "there"...]
This is the way many people say things, so students need to understand that this is the way English is, at times.
I hope this helps! (and apologies for the soapbox lecture...)