Mail armor is difficult, but not impossible, to cut. The movies do not treat armor (even plate armor) as being worthy of wear. If Hollywood is any indication, armor is tissue paper fashion. Mail has been worn for thousands of years, it existed as a form of armor predating the medieval period and was used by ancient cultures and the Romans. Even after the adoption of plate it was worn to cover the gaps of joints. Mail was experimented with in WWI and, though not for battle armor, mail is still worn and used today. Butcher's gloves are made from fine welded rings, as well as sharkbite suits. It persists because it works.

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Mail is best against cuts, especially cuts which are trying to rely on the sharpness of the edge of the weapon (think draw cuts, saber slashes, and the like.) It takes tremendous force to break through chain with the edge of a weapon, especially if the weapon isn't built with these kinds of blows in mind. A quick google search for "Chainmail test cutting" will show dozens of folks hacking against patches of mail against various surfaces, usually to little avail. Now, large axes and polearms or particularly powerful blows from longer swords designed for hacking can break through mail. Also, mail does little to prevent the bludgeoning force of a blow (in the end, even if the sword doesn't cut you, some angry man just hit you with a four pound metal stick.) Which is why mail, especially in the 11th and 12th century, was layered over and/or under padding. The way in which mail is most vulnerable is the thrust, which is why once more of the body was covered in plate and only the joints were vulnerable sword designs changed from wider cutting swords (which had some possibility of cutting mail, or at least hurting you through it) to stiffer, tapered stabbing swords (which were more nimble for getting into "chinks" of plate where the mail was) over the medieval period. If you can get the tip into a ring and break just one ring, you've opened up the mail for penetration. Couples with compression of the soft tissue, even just getting the tip of a dagger through mail in a vital area can pierce an artery.

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Spears, the thrust of swords with a fiercer point and a stiffer blade, thrusts from pole arms, dagger thrusts, arrows and quarrels were the biggest threat to mail. Again the relationship between mail and gamboised fabric (such as quilted gambesons, thick aketons, etc.) come into play, as fabric and mail react differently to different weapons. Arrows with sharp bodkin points, which are good at breaking through chain, have a harder time getting through the layers of linen which act somewhat like kevlar against them (and conversely broad cutting arrows which slice the fabric get stuck against mail like blades do.) This is where the evolution of plate comes in, as it protects better against both the cut and the thrust than mail does. The majority of wounds suffered by those wearing mail against cuts were internal bleeding and broken bones, or from thrusts. Though this is not to construe mail made someone invulnerable to cuts. The wounds suffered by the defeated at the Battle of Wisby show hundreds of wounds from cuts. Though many of them were in places the armor did not cover, such as a skull wearing a coif of mail with a cut through his nose and into his jaw (the face opening of his armor) there are wounds which cut through the mail and into the body, even into the bone. However, returning to the crux of your question, the hacking force needed to defeat the mail required more power and heft and was not "easy" the way most TV tends to show it, where effortless slices along someone's mail clad belly causes them to crumple and immediately die... Breaking through mail with a hack was a lot of work, which is why over the centuries people gravitated towards stabbing through it, which with the right tool is considerably easier than hacking it.