You have to understand the gaming principles the two dominate health systems imply (i.e. regenerating and non regenerating). In the old days, before players were able to magically mend clothing, knit shut sucking chest wounds, reset broken limbs and regain their blood supply simply by not getting shot for a bit, players were magically able to mend clothing, knit shut chest wounds etc by walking on top of a first aid kit. In those days, the supply of health went along with the supplies of ammunition as a key element of pacing and a part of the risk/reward system. Looking at games like quake 3 you find that in the ubiquitous DM_17, health and weapons are placed with a keen eye regarding this very system. Getting the mega health required being in the center of the map and spending several agonizing seconds in the air leaving you an easy target getting out with a net gain in health was difficult against a single player much less several. The railgun is placed on a platform and the only way to access it is via a jump pad, again leaving the player incredibly vulnerable (being shot in the back as you fly tends to make you fly off the edge). By forcing players to manage their health as a resource along with weapons and ammunition, confrontations become a sort of risk management scenario where you attempt to complete an encounter with a maximum amount of total resources left at the end. In the best scenario, where difficulty and pacing nearly perfectly match a player's skill the player stumbles upon health moments before they collapse and find fresh supplies of ammunition just as their weapons run dry.
With regenerating health, the focus shifts from the long term survival of the player to the short term, in the moment scenarios. Every encounter can push a player to the brink of death, which can often lead to moments of incredible fun. Notably, these games also tend to have plentiful supplies of ammunition. Sure, you can run out of ammunition for your favored weapons in Halo or Call of Duty, but it is unlikely you'll be forced to truly resort to melee attacks as your only means of defense as there are dozens of weapons scattered about for your use. Resource management becomes inconsequential and the player is instead allowed (and in many cases forced) to give a maximum effort in order to succeed.
The two approaches lead to wildly different styles of game. Health regeneration generally does away with the notion of inherent vulnerability of your character, and is thus poorly suited to games attempting to instill fear in a player (survival horror, action/horror, etc.) yet works wonderfully in games where the action of the moment is the key. Non regeneration of health on the other hand means that a player's long term survival hinges on their ability to efficiently overcome a game's challenges, which fosters a more conservative and cautious mindset. Properly implemented either scenario can lead to a fun game. Call of Duty 4's fast paced multiplayer would be tarnished if after every firefight you were forced to find health (and unless you dramatically outclass your opposition, it's unlikely you can get through many firefights unscathed). On the other hand, health management is a key aspect of Team Fortress 2's class balance and interaction, and it's removal would make the medic obsolete and the engineer's utility would be greatly diminished.