Is doubling wattage very noticeable?

It depends on where you start out.  If you're starting with 1 watt, doubling will seem louder.  If starting with 10 watts, doubling will bring the music to life and will seem much louder.  If starting with 50 watts, doubling will start to hurt....

Headroom, as tcss suggested, is the benefit...because even talking sub power, the normal listening level in db's will be lower than what you're running now.  With a well-balanced system, quality power components, and a car that is well-damped, you might just end up disappointed that you upgraded your sub's power only to find that you don't hear a difference because you just don't play it that loud. 

The key questions are:  do you feel it is lacking something now?  And would it make you feel happier to have a JBL amp?  Is the sub enclosure on the smallish side and sealed (indicates more power could probably be used)?  Or a box as large as the limits given in the specs?  (...questions are just rhetorical...)

3db is noticably louder.  About 1/3 louder than wherever you begin.  I agree, though, with the above point that the most important benefit from alarger amplifier is head room, meaning you can raise the volume farther before you begin to get amplifier clipping.

If you really want things louder, it requires 10X the amplifier power with the same driver to be twice as loud.  If you want deeper bass, build a bigger enclosure.

I thought 3db was double-volume?  Or is it just double wattage?  What db=double volume?

I agree that the new amp will sound much better even tho it won't be much louder - headroom is key with bass.