To use XCOM as an example:

The 1994 game featured a time unit stat. You would have to measure out, manually, the number of tiles you would move past, multiply that by four (the amount of TU moving one tile would take), add two if you were kneeling at the start of the turn, subtract two from the available total if you wanted to kneel at the end of the turn, and subtract from the available total the number of TU necessary for the number of shots (or reaction shots) you wanted to take. On top of that, there were three different shot types (an aimed shot that would take up almost all TUs, an auto shot that would take up about half of the available TUs, and a snap shot that'd only take up about a quarter but also maxed out at a 20% chance to hit, making it mostly useless). This process and complexity was not fun, especially since the UI didn't make it easy to count the tiles you'd be moving.

The 2012 removed TU and replaced it with a few systems. For moving, you have a pair of lines that marked a move or a dash. A basic move allows for a shot (either on-command or as a reaction), while the dash would extend range (and could be coupled with a shot based on a class-specific ability). A single character move takes seconds instead of minutes, regardless of the platform. It took an overly-complex mechanic, simplified it, and made it more fun.

Civ 4 vs Civ Revolution. Civ Rev combat is simplified in a bad way -- it's build a giant stack of combined "army" units and whittle away at your opponent, with combat resolved by die rolls and modifiers rather than unit attributes.

It's worth noting that the new XCOM was developed by creating a board game; measuring distance is common in tabletop war games (think Warhammer and its ilk).

In my mind, it's a suitable like-for-like replacement of TUs (you still have to balance movement, exposing the map, etc., with firepower) with an implementation that requires less fiddling with the UI.