There's little to no intensity to it, you just duck behind cover or whatever thing the game has you doing for a bit and then bam, it's all fine again. No intensity, no consequences.
Used to be in my day if you did shitty in a fight you had to pay for it, forcing you to play better in the next few and to, as a result, become better at the game. Oh sure it could be frustrating if you walked into a big unexpected fight with two health and a peashooter, but on the other hand if and when you actually won that fight it just felt amazing as a result. You accomplished something awesome (Insofar as anything you do in a videogame can be considered an accomplishment) and felt great as a result. Classic risk vs. reward.
Just imagine System Shock 2 or Half-Life with regenerating health, it wouldn't work. There would be no worry about those annoying headcrabs hiding in the air ducks, who cares if they nick you a bit when standing still a few seconds mends it all up? And what's the worry about scavenging and conserving?
Which isn't to say that health packs were perfect or that regenerating health is always bad. Depends on the game. But it's also easy to see why many, myself included, prefer one over the other.