It's usually tied to much, much smaller health bars.
I guess that depends on what difficulty you're playing at. I've never played a game with regenerating health that felt enhanced because of it. CoD4 on Veteran difficulty was tough because of the infinitely respawning enemies who chucked grenades with pinpoint accuracy every second. It would have felt even tougher without regenerating health.
When you have regenerating health, the only hit that matters is the one that kills you. Every other hit can simply be shrugged off because you know you'll automatically regenerate your health by simply hiding for a few seconds. This kills tension. With finite health, every hit counts. You never know when you'll next get an opportunity to heal so you have to avoid getting hit whenever possible.
The implementation of finite health systems obviously has an impact on the amount of tension it creates. If health kits are too abundant, tension is lowered. If health kits are too rare, frustration likely ensues. None of these are shortcomings of finite health systems, though. You mentioned that Deus Ex was too easy because of finite health. No, it was too easy because the game paused while you accessed your inventory, medkits were too common and they healed instantly rather than over time. These are problems with implementation, not the concept itself. It's a tricky thing to create a balanced finite health system, which is why most designers these days don't bother. Regenerating health is just inherently easier for designers and players alike.