Double XP will cause your "Player Level" (Your actual level) to progress at twice the normal rate, which will cause you to unlock new guns and perks at twice the normal rate (while it's active)
Double Weapon XP will cause your "Weapon Levels" (the level noted individually for every weapon) to progress at twice the normal rate, which will cause you to unlock new attachments for that specific gun at twice the normal rate (while it's active).
Double XP will not cause your guns to level up faster, and Double Weapon XP will not cause you to level up faster.
Both can be active at the same time however, causing both you and your guns to level up twice as quickly.
gaming.stackexchange.com/users/12087/Mr Smooth
Sorry to answer my own question...
but... After searching more, it sounds like you can unlock things specific to weapons such as attachments, etc with "Double Weapon XP", but it doesn't help your overall XP.
Similarly "Double XP" helps your overall XP, but not weapon-specific XP.
I found the answer here: http://www.gamefaqs.com/boards/995751-call-of-duty-modern-warfare-3/60609664
gaming.stackexchange.com/users/5866/Andy Fleming
Weapon xp inadvertently affects your regular xp because as your guns level up faster you reach mastery level (a challenge) which gives you 10,000xp....
which helps level you up.
gaming.stackexchange.com/users/21374/hello
Double XP will cause you to level up faster, unlocking new weapons faster etc.
Double Weapon XP will only cause you to get closer to unlocking emblems or callsigns.
gaming.stackexchange.com/users/24322/unknown
Double xp makes you rank up faster example: reaching commander faster Double weapon xp makes your guns rank up faster example:unlocking attachments,proifencies,camo,reticle faster
gaming.stackexchange.com/users/33628/trickster
What's the difference between "Double XP" vs "Double Weapon XP" in Call of Duty Modern Warfare 3?  Which is the better choice?
"Double XP" vs "Double Weapon XP" in MW3
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<call-of-duty-modern-warfare-3>
gaming.stackexchange.com/questions/34590
gaming.stackexchange.com/users/5866/Andy Fleming
Double XP wil cause your player level and weapons level to progress at twice the rate.  It is apparent that each option has its own perks. There seems to be no better choice, however, since both can be active simultaneously. 
The  advantages and disadvantages are being described.