TSR D&D--especially 1E AD&D--had a bunch of things that, by today's standards, made the game more deadly, especially at low levels. For instance, magic-user HP (d4 at 1st level), certain undead draining levels, no healing surges or HD, no death saves, etc. A lot of groups bypassed this stuff, or house-ruled it out.
"Old school" isn't really a specifically defined thing, but includes a bunch of qualities, all of which are part of D&D today, but the OSR just emphasizes them more, and excludes a lot of newer stuff that came out through WotC and other more recent influences: Warcraft, anime, etc.
Any version of D&D is ultimately malleable. Some, like myself, prefer the 5E rules set, but use OSR(esque) stuff for inspiration and adventuring. Others are diehard "OSR or die" types, who won't touch anything published by WotC.
There's also the cultural aspect of the OSR which, quite frankly, tends towards machismo. I only mention this because such folks tend to like a more survivalist play-style. But this isn't intrinsic--or at least inviolable--in playing an "old school flavored" game. One could play 5E in the Wilderlands or Hyboria, focus on hexcrawling and minimize world-shaking events.
I think, also, that there's a strong correlation between:
OSR and sword & sorcery fantasy
New school (WotC D&D) and epic fantasy
Sword & sorcery fantasy was the dominant mode, at least of commercial fantasy, until Tolkien came along, and gradually petered out, especially with the rise of genre fantasy in the late 70s and 80s, but existed as a kind of secondary stream, morphing into (or giving birth to) grimdark in the late 90s and early 00s with George RR Martin and his "children": Joe Abercrombie, Mark Lawrence, etc.
Gary Gygax is to the OSR somewhat what Robert E Howard is to contemporary grimdark. They aren't the same thing, but the latter are in the "spirit of" the progenitor. But there are different ways that this spirit can be honored, and different degrees that it can be incorporated into a campaign.