While not particularly good at absorbing an impact wound from a blunt weapon like a mace, mail is actually very good at limiting or preventing lacerations from a sword blade or the like. Given that typical sword (of the european variety, at least- less familiar with the particulars of katanas) weighed only a few pounds and had a balance point very near the hilt (to keep the blade maneuverable for better control), the actual energy delivered through the impact of the blade against mail is pretty minor, particularly by comparison to an axe or a mace. The main objective of a sword is to slash or cut (or thrust/stab, of course), and mail is ideal for deflecting such cuts.

I'll let someone else elaborate on its advantages with regards to arrows, as I don't know a tremendous amount about the physics behind arrow-to-mail impact. At the very least though it would be better than having nothing between the arrowhead and your flesh- it's another (relatively sturdy) layer for an arrow to penetrate.

Mail is not -great- protection from piercing weapons, but I think it's weakness can be overstated. All examples of mail(le) that I know of from Europe are riveted, meaning that each ring is riveted shut. So those rings are much harder to pry open than ones that are just closed with pliars.

Source: Arms and Armour of the medieval Knight, Edge and Paddock.

Footnote: at no time did medieval people refer to mail as 'chainmail' or 'chain', and at no time did they use the word 'mail' to modify other types of armour like plate or scale.

Source: Claude Blair, European Armour circa 1066 to 1700.