Hard to say. But I'd bet that Linux's memory management is more efficient.
Keep in mind that efficient memory management means:
"use as much main memory as possible and provide the most effective way to provide memory space to applications for fastest operation".
Or something like that. In other words, unused memory is wasted memory.
It's very hard to tell unless you do extensive benchmarks though and there are other things that are fast with Linux such as I/O scedualers and such that are superior to windows, more then likely. And out of those there are several to choose from depending on your application.
Other things aren't so hot for Linux.. for instance if you use a Gnome desktop enviroment when you start applications they do a lot of polling for files that may or may not exist.. and all this wastes a lot of time compared to Windows applications starting up. Windows has the registry, which is fast.. something like OpenOffice.org may spend 2-3 seconds just searching around the harddrive for files when it starts up. Also something like that is bloated compared to something commonly used in Windows like Word.
Microsoft spends a lot of time making sure that everything is 'snappy'. When you press a button to start a program, or do something.. they make sure something happens right away. They pre-load application data into memory so that it's already loaded when you start things like IE or MS Office. IE itself is just mostly a shell that calls system libraries to build a application, and most of that data is already used when your desktop and taskbar and such start up. It doesn't matter so much that it is usable quicker, or that it does something usefull right away.. it matters that whatever happens first, happens quickly.
Ultimately it may take something longer to _finish_, but the important part is that it starts quicker.
With Linux application designers and such are not so sophisticated at doing stuff like that. When you move windows in Linux, they tear. When you hit the application menu for the first time after a boot it may take a half of a second to do something. When you launch a application you may see nothing for a few moments which makes you think that maybe it didn't do anything for a moment, which increases the perception of slowness.
So to get a 'feel' for the difference is a very difficult thing. What you end up experiencing goes thru several layers of software code in either operating system. Very subjective stuff.
Also keep in mind that Linux has had four-five years of developement since Microsoft last released a desktop OS. Linux 2.6 series memory management and scedualing system is state of the art and very quick compared to other (contemporary) operating systems and has had extensive developement work on it from companies like IBM and SGI, which are very good at this sort of thing.
Also file systems are in a high state of tune. One that is considured relatively slow, ext3, is still going to end up being much faster then NTFS for many things.
And I/O speed difference will mask many differences in memory management. Maps may load up quicker, games start faster.. but the FPS (which is dependant on hardware instead of software so much because it's a single task, single threaded thing most of the time) is going to be pretty close to the same, unless you have a ATI card, in which case it will suck.