To start, there are cosmetic differences: Minimize does a little graphic effect and puts a shrunken icon of the window into your Dock (on the right hand side, not with the apps, although this can be changed). Hide makes all of the application's windows simply disappear from view without any changes to the Dock.
But the big difference lies in how you make the windows reappear.
Minimized windows require a mouse click in the Dock to summon them.
Hidden apps will reappear if you Command-Tab to the app, so you can bring back the windows using the keyboard only.
(You can also click in the Dock to get a hidden app back.)
Because I use Command-Tab heavily, and try to avoid the mouse, I use "Hide" quite a bit but never use "Mimimize" as it requires mouse interaction to get the window back.
As others have noted, you can set the Dock preferences to "minimize into application icon". This doesn't change the keyboard/mouse equation; all it does is prevent a "minimized" window from getting its own icon in the Dock - instead you have to click on the app icon in the dock to un-minimize the window.