If there are similarities between the new and the old schools of gaming, there are differences too. One of the biggest is in their approaches to genre simulation.
In order to emulate a genre in the old school one frequently takes a set of rules and alters the skills and equipment. FGU's Bushido and Daredevil take the same central mechanic to very different places. Likewise, RuneQuest, Call of Cthulhu, and SuperWorld are each about as different from each other as imaginable, yet the core system remains.
This approach works, and it hasn't gone away. FFG's Rogue Trader, Dark Heresy, and Only War are each slices of GW's grimdark 40K world, but the system heralds back over two decades to Warhammer FRPG.
In the new school, in order to emulate a genre one tends to focus on what the genre does, and designing mechanics specifically around that. Skill and equipment lists may still exist, but the entire game focuses on the goal of play. Unlike the older games, the narrower the focus of the play style the more able the designer is to craft mechanics to support it. The games, seen from an old school perspective, tend to lose versatility.
Not that new school games can't be re-skinned and re-purposed. Apocalypse World gives us Dungeon World and Monster Hearts, for instance.
I think the crux of the great divide between the two approaches lies in there, somewhere. To people used to the older style, new school games seem so narrowly focused. It's not a Western, or a Supernatural Western, it's a game where you play wandering agents of the religious authority, dealing with problems of the Faithful on the Frontier. The task system deals with faith, with belief, not with how many miles an hour one can ride a horse. There may not even be any rules for horses, horses aren't central to the theme and play style of the game. To someone steeped in old school games this seems a glaring oversight on any game set int he old West.
The reverse is true as well. To someone used to new school games, the rules of the old school games often seem vague and directionless. What do the rules tell you about the presumed style of play? How does the texture match the flavour? Why does the game need twenty-seven kinds of straight one-handed swords?
There is new vs old, but underlying that is focus against versatility.