The map_heights.hgt file contains basically the same information as the map_heights.tga pre-converted into a format the game recognises. It allows more variation in height increments than the map_heights.tga as it uses float decimal values instead of just the 0-255 values possible using the RGB values in the tga.
The map_heights.hgt file is not strictly required for most versions of Rome or Medieval II - Total War. The file does not re-generate on its own if it is deleted; unlike the map.rwm file. A new version of the heights.hgt can be made using Alpaca's TGA-HGT Converter and via IWTE.
A map_heights.hgt file is required for RTW /BI versions 1.3 and 1.4 - however these versions are largely superseded by patches 1.5 and .1.6.
Minor amendments have to be made to the Medieval II - Total War map to allow it to load without the heights.hgt. There is currently some debate as to whether having a heights.hgt for that game improves the appearance of the coastline as it allows more widespread use of 'low land' with RGB near 0,0,0.