I think I'm in agreement with most of what has been posted so far.
It may not be a popular opinion, but despite seeing
Abre Los Ojos (1997) first, and originally insisting that
Vanilla Sky (2001) was a bad Hollywood remake: I eventually came around to preferring
Vanilla Sky after re-watching them both a decade later.
But it's been another decade since then (just writing that hurts) and my opinion may change again if I were to watch them now.
Though I don't know that I'd call the remake
better, I remember thinking that both
Girl with the Dragon Tattoo movies were good in their own right.
Heat (1995) was apparently a remake of Michael Mann's own made-for-TV movie
L.A. Takedown (1989) - and though I've never seen it, I can't imagine that being better.
I haven't seen
The Departed (2006) since it was released - so I can't say if I like it more, but I was a bit disappointed with
Infernal Affairs (2002) after finally watching it recently.
Hannibal did better adapting Red Dragon, Hannibal and slightly Hannibal Rising than any of the films. Especially the Hannibal novel.
If they ever would of gotten the rights to Silence of the Lambs that would of been the real test.
I remember thinking
Manhunter (1986) was better than anything which came after it; but I did not watch much of the
Hannibal TV show.
I'm always hesitant to recommend any media if it's not fresh in my mind - especially older media, but it might be worth seeking that out if you haven't seen it.
I really liked the 12 Monkeys show too.
The Terry Gilliam movie
12 Monkeys (1995) might also be considered a remake, as it was inspired by the short
La Jetée (1962).