Depends how how many games you play and do you download downloadable content (DLC).
For example, I have an older XBox 360 with just 20GB, but I get by because I usually just play one game at a time. It is recommended to "install" your games to the HD for faster load times and less noise, so that requires about 6-8GB.
Also, do you like to try out game demos? Those are usually a couple of GBs. Do you plan on buying a bunch of XBox downloadable games? The XBox arcade games are usually under 1GB each.
Do you plan on downloading movies and music to your hard drive?
If you answer no you most of these, you can even go the barebones route and get an XBox without a HD (arcade model) and just buy a 16GB flash drive to store your content.
If you answer yes to a lot of this stuff, then bigger is better.
But it really depends on how many games you wish to have access to on the hard drive (demos, installed games, DLC).
BTW, you might want to check out the newest model Xbox called the XBox 360 Slim. It only cames in one HD size, so it makes this question moot.
First of all, keep in mind that on the 360, saved games and Xbox Live profiles are ultimately stored in different "places". While you can keep both on a hard drive (or on a memory unit or flash drive if the saved games are small enough), saved games are only kept locally, while your Xbox Live profile is tracked on Microsoft's servers.
If you're talking about an Xbox Live profile, then it will "move" the first time you recover it - once you recover it from Microsoft's servers, that becomes your up-to-date profile, so you won't need to copy it from your hard drive. Your saved games won't be accessible until you get that enclosure so you can connect your old hard drive.
Local profiles, ones not associated with an Xbox Live account, are only stored locally, so if that's what you have, then you'll need to wait for the enclosure to arrive.
Once the hard drive is connected to your new 360, you can move whatever saved files will fit to the internal flash drive. (If you go to My Xbox, System Settings on the far right, and choose Memory, it'll show up as Memory Unit (3.x GB free), and you can copy files from your hard drive to it and vice versa.) I don't personally believe there's a difference between storing your profile on the internal drive and storing it on the hard drive. However, if you find yourself playing away from home on occasion, you may be better off picking up a small USB flash drive (up to 16 GB), formatting it for the 360, and storing your profile on that. You can simply pull the USB drive and take it to your friend's house without needing to recover your profile at either place.
Best Answer
You should not have problems. Since it is an HD that is "Microsoft Aproved", you are good to go.
There are some tutorials at the internet where some people are even using the old Xbox 360 HD inside Xbox 360 Slim version. Is just a matter of removing this drive from inside the "fast eject" case Microsoft created for its hard drives.
You can swap hard drivers on your Xbox devices, as long as they are using Microsoft certified hardware.