If I'm logged into my computer locally using User "A", then go to
another computer and login to my computer using REmote Desktop, using
User "A" again, then get back on my computer and login locally, using
the same (You guessed it) User "A", all my apps have been closed and a
new session started.

Let me add this:
My computer is on a Windows Domain, and I log in using "A", when I log
in using Remote Desktop, "A" probably isn't really the same user,
because in this case its the local computers account (I would guess).

How can I keep my session from being killed when I log back in as
myself?

Wouldn't you know it... I search and search and have no answers, and 1
minute after posting my question I find the answer!

I wasn't part of the local Administrators group!

That did it.