You're right, the Magma Axe is a touch better (higher average damage, higher max damage, higher critical - all marginal though, and has the rare potential to hit for fractionally less than the sword). There aren't many Hand Axes in this game and they're all comically undervalued to their Longsword equivalents. I mean, even the lowly Sword of Flames costs more than the mighty Magma Axe!
Axes generally are a class without purpose. The Bashers (Hammer/Mace/Maul) have utility as Clerics can use so few other weapons. Shortswords and Daggers (both light) differentiate themselves in that Clerics and Mages can use Daggers and there are so many great Rogue-only Daggers (including the best overall Dagger and the best mid-game Daggers). But the only reason to have the Axe class is for flavor - not utility. The Hand Axes? Warriors and Rogues but both of them can use Longswords. The Greataxes are Warrior only but they get Greatswords anyway. That's not to say there are no good Axes, there are some great ones esp the 2H ones, and as said they're not identical to Longswords, but they're for flavor not need. Was the Thief class to prohibited from Longswords, or if there were Rogue "Ranger" skills that boosted axes, or if they had a different attack speed, they might have their own real niche. But as I say the only reason they exist here is for RPG flavor not to have a unique role in game mechanics. [And btw, nothing wrong with stuff only for RPG flavor!]
PS, the one that always bakes my noodle is the janky Bluesteel Greataxe is worth a lot more money than the better Minotaur Greataxe and the MASSIVELY more powerful Axe of the Depths and Permafrost Greataxe. There are some strange prices throughout the game; Troll Cleaver and Stiletto are two of the best, unique, weapons kicking around and they're worth about as much as steel Longsword LOL.