Using points as currency to buy units in wargames has been popular since at least the 1970s, when a point-based design-your-own system for the Panzerblitz tactical armor game appeared in the pages of Avalon Hill's The General magazine.
That system was based on a simple formula, but sometimes point values are assigned by a more subjective process through playtesting. Probably most common is a mixture that starts with a formula and then tweaks the values based on playtesting experience. Most systems base their values purely on game impact, but sometimes the point value of a unit is boosted (or more rarely discounted) because of the unit's rarity on the battlefield. Advanced Squad Leader is probably the most well-known example of this approach.
The primary alternative to a point-based system is one that attempts to make all units of roughly equal value and carefully balances the game impact of the units accordingly. Some examples of this approach include Ogre, Victory and Navia Dratp. This naturally imposes some limits on the power of the pieces and is pretty intensive from a playtesting standpoint.
The advantages of a point-based system is that it can be wide open, with units of widely varying strength. It can include goblins and dragons, PT boats and Battleships, jeeps and King tiger tanks in the same system.
This can also be a weakness of point-based systems, however, because not all points are created equal. In some games, for example, the combat values or the rules may make a certain kind of unit virtually invulnerable to other kinds. In War at Sea, for example, 48 points of battleship (say the HMS Hood) is not worth as much as 48 points of U-66 class U-Boats. In fact, the battleship is doomed unless it has some supporting destroyers or aircraft because it has no weapons that can hurt the submarines. Point-based systems are vulnerable to mismatches of this sort or "degenerate" strategies that rely on extreme builds. Point-based systems work best when bounded by some kind of restrictions on the types of units built. For example, War at Sea mitigates the potential abuse of an all-sub or all-aircraft fleet by allowing only surface ships to claim objectives. In the Dungeons & Dragons Miniatures game and in Axis & Allies (land) Miniatures there are limits on how many pieces can be in an army, so a player can't overwhelm his opponent with a horde of very-low value pieces.
Still, I read occasional complaints on the Axis & Allies message boards and elsewhere about the point values assigned to various units. Commonly posters argue that XX points of units should have an equal game effect. Ideally they should, except that in a game with as complex interrelationships as Axis & Allies not all points will be created equal, especially across unit types. I would agree that battleships of the same point value should have similar game value, and that subs of similar point values likewise. But it's unreasonable to expect that 50 points of subs will necessarily have the same impact as 50 points of battleships without considering the impact of the opponent's builds. Accounting for the differences would probably require an excessive amount of complexity.
Likewise, some complain that some nationalities are too powerful or too weak, based on the point system. With a history-based game such as War at Sea or Axis & Allies Miniatures there are limits on how much the game can even things up. Japanese and Italian tanks sucked and there's only so much that can be done to help them out. American warships had lots of AA guns and Japanese Long Lance torpedoes were an incredible weapon, there's only so much that the game can do to mitigate those facts without doing violence to the history.
Points are useful, but competitive players should be mindful of the limits of a points-based system and consider including other limits as well so that the playing experience isn't spoiled by uncompetitive builds.