This one is maybe the closest in a tourney filled with close calls.
Both are masterclasses in all the aspects that make for great popcorn entertainment with both depth and visceral thrills. Ripley and Mclane are both relatable protagonists surrounded by a supporting cast that is always engaging and interesting. So not only is the "head" of these films more than solid, the "guts" are too. Both directors get the best out of their respective casts and they leverage the production values of the time to the hilt. Neither film has any kind of glaring or outright terrible flaws.
So... A choice is gonna come down to a lot of personal feelings... And given how intimately these films are woven into my youth... Even then this will be a squeeker... But if I have to make a choice...
Aliens.
Sure, it is a sequel and maybe that should warrant a demerit in some eyes as compared to the original film that is Die Hard. And it is not quite in the same category given Die Hard is a pure action film and Aliens is a sci fi actioner.
Still... I guess I give Cameron's film the ever so slight edge on a lot of aspects that Die Hard also does well. How much of an underdog are the protagonists? Sure, John is barefoot, isolated in a skyscaper against a crack team of mercinaries led by a mastermind with both his and the lives of his wife and her co-workers on the line. Great stakes. But Ripley is dealing with her own traumas and fears on top of being marooned on a planet now infested by hundreds of her worst nightmares who will overrun the weak defenses she and the Colonial Marines have set up, but on top of that they are all on a deadline of having to do something before the power source of the atmosphere generators goes critical and explodes in a nuclear fireball. Gotta go Aliens in terms of stakes and urgency.
While both main characters are great, and this might sound strange, while Ripley is a bit "flatter" than John I actually count that against Die Hard. Mclane is a fantastic character and for what DH is going for he's perfect. But part of what is done with him is to inject humor into the film and to cut the tension from time to time to give the audience a break, some breathing room. And that was done incredibly well. That's not a complaint. End of the day though I just prefer Ripley's characterization more. Unlike with John what is done with her is not about pausing the story for a laugh. While DH is indeed very relentless in its pacing Aliens is just a bit more so. Ripley doesn't have time to drop any quips or one liners for most of the film's run time until the third act finale, and that seems to be because Cameron wanted maximum impact. Mclane is indeed more quotable but Ripley never takes you out of the moment.
Both films are amazing works of production values. I mean we have incredible sets, pyrotechnics, SFX, costuming, make up... All the tricks of the trade being employed in both movies. Everything feels real. But again... Gotta give Aliens the edge. It might be unfair since as a sci fi film in a sci fi setting of course it requires more out of the production team than say, DH. But the world Cameron makes is just undeniable. Not just the creature effects but the environments in every section of the film.
As for on screen action... Again this is a hard decision. John is not Arnold or Sly. He is not some unstoppable killing machine and each of his fights take a toll on him until he is roughed up and near broken by the time he meets Hans at the end. John is one man against many and the action is always varied and filled with tension. That said... Aliens does much the same but bigger and with more characters. Ripley and the Marines are always on the short end of the stick, always on the run, never in control of the situation. Your fears for them are always out in front. This isn't a story of triumph. This is a story of pure survival.
I could go on and on... Both of these films are Ur-texts of blockbuster film making and have been the wellspring for movies good and bad though I dare say neither have been equaled. I give the edge, and just that, the edge to Aliens over DH. I love both dearly but in the end I had to give this decision to Cameron's amazing sequel.