Reading through various old-school games, one of the biggest rules differences that stands out to me is the concept of race-as-class. With my start in D&D 3E, I rapidly got used to the idea that a character of any race could be essentially anything: elven fighters, dwarven wizards, half-orc clerics, halfling barbarians, and so on. These characters might not be completely optimized, but they'd at least be playable (aside from the Linear Fighter, Quadratic Wizard issue, which is a different blog post entirely).
Not quite so in old-school games. Dwarves, for example, are pretty much prohibited by the rules from learning any type of spell, as dwarves are generally only permitted to advance as fighters or thieves. Similarly, elves are typically only permitted to advance as fighters and magic-users.
This seems really weird, because in later editions, it seems to me like it became much easier to play characters that broke the typical archetype mold that early D&D and a lot of the OSR systems push characters toward. Dwarves and elves (and other races) became much more varied in later editions, where as the earlier games seem like they tended to push the fantasy stereotypes, for lack of a better word. Dwarves and elves were almost all somewhat like Tolkien races; elves were magical and ethereal, dwarves were sturdy and tough, halflings were jolly and dextrous.
This isn't to say this is a bad thing. Archetypes are archetypes for a reason: they're classic, they immediately show what a race's strong points are, and they're a good starting point for new players. They're also quick to explain. "I'm playing a dwarf who uses an axe" is a lot easier for a new gamer to parse than "I'm playing a dwarf warlock that can use magic and has a sword," simply because "dwarf with an axe" is immediately recognizable as very similar to common dwarves in movies or video games, like Gimli from Lord of the Rings.
Now, separating race and class shouldn't be too hard to house-rule into OSR systems. It's a matter of giving humans some kind of bonus or ability to balance them out with the extra abilities of the other races and removing the "advances as X" bits of the other races. This would add some customization options for characters, albeit at the cost of also adding some complexity. Whether that complexity and customization is a good fit for the game is up to individual GMs and their groups.