Is there a reason to increase them beyond level 39 (enchanting and anvil at maximum)? Does the armor rating increase?
I noticed (or read somewhere) that the level is also decisive for the prices you get trading with villagers, especially with the "Hero of the Village" perk.
Please note: I considered the wiki on experience and it seems outdated.
Other than bragging rights, there is no benefit to hoarding experience. Experience converts directly into levels, and levels are used as a currency when paying for enchantments and anvil use. And since each level increase requires exponentially more XP, you'll need far less experience if you don't hoard it - for example, it will take vastly less XP to perform ten enchants by repeatedly leveling up to level 30 and immediately spending the experience on a high grade enchantment (dropping you to level 27) ten times, than leveling to 60 then spending 30 levels in one enchanting session.
Your current level or amount of experience has no impact on gameplay; it's a resource to be spent, not a player stat.
In particular, experience can be farmed, and XP farms are some of the most popular farms in the game, allowing unlimited use of anvil (sans durability), enchanting, and repair of equipment with the mending enchantment (repair applied only at the moment XP orbs are collected, can't use accumulated XP).
You might have been confused by the phrasing of:
Level I Hero of the Village decreases the cost of the first item in a trade by 30%; each additional level decreases the price by an additional... [source: wiki].
It talks about the effect level (similarly to "Sharpness I, Sharpness V") and not the player level.
