Ummm, you're forgetting one very big thing and a smaller, though equally important thing.
1) You have a brand new OS installed - it's going to be snappier simply because it does not have all of the junk in the registry yet that accumulates over time from regular every day use of a computer that is Windows based. If you talk to the folks like me who maintain numerous computers in networks, you'll see that we have systems for reverting them back to a default stock system - imaging software, things like DeepFreeze, etc. which take the computer back to a stock image over a period of time.
Folks like us also do the same thing to our personal computers (especially me, since I beta test a lot of applications). Every 6 months to a year we'll format the system drive, reinstall the OS, and install the current version(s) of apps that we always have installed - for a much smoother experience until things start to slow down (inevitably) once again.
2) The smaller, though equal thing to consider, is that, while a 32bit OS says that it can address 4 GB of RAM, in actuality, it only addresses about 3.0 to 3.3 GB of your 4 GB, reserving the rest for system use that you'll never see. Your 64bit OS will address all of that 4 GB without the reservation at all. But, all in all, an increase of ~750 MB of ram is not going to make as noticeable difference in your computer when going from ~3.25 to 4 GB as it would if you were on XP and originally had 256 MB and boosted that to 1 GB. Windows 7 is by far the most efficient of all Windows OSs in terms of memory management, and at this level it is not all that different.
_________________________________
Now, that being said, I'm running 12 GB (DDR3) on my machine because I like to multi-task - a lot. Between Firefox and Thunderbird alone, I'm usually eating up 3 GB of RAM to begin with, so....doubling your RAM will make things operate much smoother, but will not necessarily make things faster - the program loading off your HD will still load the same, but having more RAM for it to stay in memory as opposed to having to swap to the HD will definitely improve efficiency and make it seem snappier - but going from almost 4GB to a true 4 GB will not be as much of the cause of the snappiness for you right now as the fact that your OS is a brand new install.