On my PC (Vista), with a 21" CRT monitor (1600x1200), HoMM3 scales perfectly in full-screen mode.
On my MacBook Pro (Windows XP, 1400x900), however, it runs in a box in the middle of the screen ("windowboxed").
Why is that? How can I make it run on the MacBook like it does on the PC?

If you find the game folder does it have a conf file you can open with notepad? If so you might beable to change some settings in that to get it to run on the screen fully. If you can't work out what to do just post the contents of the conf on here.

No, I don't think there are any such files in this version of HoHMM.

I also didn't find a way to do this via a config file. I can't find one anywhere.
On my 24'' lcd (1920x1200) the game also runs in a box in the middle of the screen, so i'm guessing the only way to run it in full screen is in 1600x1200 (like your CRT monitor) or 4:3 equivalents.
Our resolutions, 1400x900 and 1920x1200 are 8:5 (i think) and i guess that's the problem.