They're definitely thrillers, but I wouldn't go so far as to label them psychological thrillers, cause that's putting them in the same pool as Serial Experiments Lain.
Monster is the better series and is higher on my personal list, but Death Note was the first anime I ever watched in full, so I'll always have a special place for it in my heart.
Monster is consistently ominous and that stems mainly from Johan being mostly secluded to the shadows, making his appearances that much more intense. Death Note actually focuses on the antagonist and what he's thinking at all times, which kinda works against it in that regard. It's certainly more intense where the cat-and-mouse aspect is concerned, but it's an especially grim storyline, at least all the way through. Not to mention it has a lot more lighthearted moments than Monster ever did.
As for characters, Tenma and Nina's journey into darkness and Johan's sheer brilliance as a villain kick it up a notch, but I only consider Johan an all-timer. Light and L are both top contenders where best anime characters are concerned and Light's journey as a character is pretty much unrivalled, plus a good deal of their team have varying levels of complexity to them, as well, so I give this one to Death Note.
Light and Johan is a tough one. As a well-rounded character, Light's almost par-none, but Johan is just so good as a villain that it makes the virtual lack of reasoning for his evilness acceptable. I give to Johan any day.
Lunge was a great secondary antagonist and I really liked his characterization, but c'mon, how can you not love L? No freakin' contest. lol
