Actually its addressing space.
32bit allows 4Gb of address space.
So, in a nut shell (assuming 4Gb are installed) its 4GB minus Video memory = approx amount of RAM useable by the OS.
To hopefully better explain:
Lets assume you have a dedicated graphics card that has 512MB RAM.
With a 32bit system, you have 4Gb of address space.
This means that you have 4GB of space to map everything to. (RAM, Video cards etc).
So, 4GB RAM + 512MB Graphics Card= 4.5 GB total adress space.
The system doesnt not have the room to address it all.
So it will address all periphials first, in this case a 512MB GPU, then assign what space is left to map physical RAM, leaving you a useable 3.5GB approx out of 4GB installed.
With a 64bit OS, that address space is significantly more as DocBrown has shown above.
So the system is able to address the 4GB of physical RAM installed, as well as your dedicated graphics card on board RAM (in this case 512MB) leaving you 4GB of useable system memory out 4GB installed.
And a total of approx. 4.5GB of address space assigned.
However, as DocBrown also pointed out, you will still have less 4GB of useable RAM however with a 64bit OS.
The reason is, onboard video does not have its own RAM.
So it must borrow memory form somewhere. So it does so from the installed system RAM at whatever value it is set at in the bios.
In your case, it is not an issue of needing more addressing space to assign everything.
The integrated GPU itself must borrow its RAM from the system itself.
Even with a 64bit OS, you will still have less RAm useable than is installed in the system.
With a integrated GPU, how a 64bit system can help is amount of RAm it can adress.
So if you were to put 8Gb into a 32bit OS, youd have the useable RAM you do now.
If under 64 bit, it would raise, minus the set value your integrated GPU uses (if set to 512MB it would show 7.5GB useable out of 8GB)