IMO Shadow Warrior 2013, while a good game, it doesn't really feel like an oldschool game. There is little meaningful exploration - sure you can collect trinkets and stuff, but it isn't the same like looking for a better weapon, powerups, etc in secret areas and the skill system took away from it (instead of exploring the level to find stuff, you kill the enemies to collect XP and upgrade yourself). I think
TotalBiscuit's video about oldschool FPS design is spot-on (which also uses the classic Shadow Warrior as an example). Also Shadow Warrior 2013 seemed to have very little vertical variation - even the classic Shadow Warrior with the limited Build engine had more vertical variation. As TB says (or i think he says since it has been a while since i saw that video :-P), a big deal with these old games was mastering the level. The level isn't just a backdrop for you to kill baddies, but it is an entity by itself. Personally i'd add that navigating and exploring a level should be entertaining by itself (of course just the level cannot stand by itself, except in games where exploring the environment is the main reason people play, like in Stanley's Parable).
When it comes to level mechanics i found the Rise of the Triad remake much better and true to the classic FPS. My two biggest problems with the new ROTT however are the lack of polish (several little bugs, especially movement bugs - loosing your "flow" because a little polygon got in your way is not fun, the levels should use more invisible collision geometry) and the level design/layout doesn't seem very well thought out. It may remind a bit of the original ROTT, but the original ROTT's engine limitation made things a bit more straightforward. Also the levels had many areas where looked a lot the same and there wasn't a minimap available to help you out. Classic FPS games had minimaps which helped exploration, but few modern games have them and when they do they're often very crude top-down approximations. I think the last FPS game to have a proper minimap based on the level's layout (which was fully 3D) was Dark Forces 2.
I think level design-wise the new Shadow Warrior was better than the new ROTT, except the lack of vertical variation. However its mechanics weren't very different from Flying Wild Hog developers' previous games, like Hard Reset, Painkiller, etc and all of them had a problem with too simple arena-like layouts. The Shadow Warrior levels are an improvement, but IMO not a big one and certainly not like the FPS games of older times.