More RAM will allow you to create larger (in terms of polygon count and texture size) scenes.
It might help some with render times; but for the most part, render time is [almost] entirely dependant on CPU power. There is also some minor fluctuations depending of HDD speed, RAM speed, stuff like that.
The major decrease in rendertime for having more RAM comes from having to access pagefile memory (which is much slower than RAM) less often.
The 'hardware assisted render' setting can use GPU (video card) cycles for part of the rendering process.

32-bit is 2^32, or 4 Gigabytes. Subtract reserved system (hardware drivers, OS req., and so forth) requirements and you get the total amount of physical free RAM. Every running program, even background processes, uses this free RAM; so subtract those req. also.
Generally, starting w/ 4GB, you end up in the 2.5 to 3.5 GB range.
Futhermore, unless you tamper with your boot.ini file; no 32-bit process will use more than 2 GB RAM each.
A 64-bit OS can theoretically 'see' 16 exabytes of RAM (~17 billion GB). Current 64-bit OS are capped at something far below that (128 GB for XP pro or Vistas above HP), which is still a lot more than you are going to be able to get into any existing PC motherboard.

If you have both a 64-bit OS and a 64-bit application, you can use up to the artificial OS RAM cap.
Example would be Vue 6 Infinite and 64-bit XP Pro being used together. You could then access and use up to 128 GB of RAM, minus system requirements, on a single scene.

Friends don't let friends use booleans.