The Health Bar is more realistic than Regenerating.

[Baby Tea]

So enjoying the game has to come at the expense of difficulty and challenge?

The challenge and difficulty is only gone if the developers take 'health pack' style levels and AI and throw in regenerating health. Regenerating health opens the door for better AI and more challenging levels. Since players make their own health, the game needs to respond, challenge wise, accordingly. If it doesn't, that's the fault of the developer, not the mechanic.

Games with regenerating health, like Rainbow Six Vegas, can be plenty challenging.
Regenerating health only removes challenge if the developers didn't balance the game properly.

 

[KalosCast]

It's good for competitive multiplayer to either have regenerating life (a la Call of Duty and Halo) or to have healing items be quick and plentiful (Battlefield series). This is to prevent instant-lose scenarios where simply having to fight a second person before you got a chance to get to the medkit spawn means that you're virtually guaranteed to lose the fight, regardless of personal skill. There are a few games that are good without regenerating health, such as Left4Dead and most MMORPGs, but this is because resource management is a core part of strategy in these games (one could argue that MMORPGs are entirely resource-management games thinly disguised as RPGs), they're relatively slower-paced (in the grand scheme of things) and you have more defensive options beyond "shoot the other guy harder than he's shooting you."

As for single-player, I think something like the first Halo game is the best of both worlds. The regenerating shield keeps things moving and ensures that the game will never become unwinnable simply because you made one mistake, while at the same time doesn't let you come back to full mere seconds of being on the verge of death.