I have used Wine before. I recently heard of VirtualBox. Do they do the same thing? What are the differences and relative merits of these?
They are not the same, no. VirtualBox is a "virtual machine", which means that it create a system where the software inside thinks it is on a real piece of hardware; VirtualBox can run Windows, MacOSX, Linux, SunOS (for x86), etc. It would be a operating system once you start and then you would need to install the applications you desire to run. With VirtualBox, a Windows app will look like a Windows app and a MacOSX app will look like a MacOSX app.
Wine is a MS-Windows interface emulator. It mimics the windowing libraries so a MS-Windows GUI program can display in XWindows instead of WinXP/Vista/Win7. It is not an environment, it can only run one program (but you can start multiple wine apps). Because it is mimicing the standard MS libraries, not all Windows programs may run under it if they need additional libraries or if they bypass the standard libraries and try to access lower-level libraries/interfaces. With Wine, it may not look like a Windows app once it is running.
VirtualBox takes a lot more room (creating a copy of the guest OS), but it is much more reliable than apps using Wine (usually not Wine's fault, but the apps fault). And if you have an application that needs support programs (like Putty using Pageant), that won't work with Wine, but works very well in VirtualBox.