I like Monster a bit better but in the end they're actually quite different..
In Death Note you'll be following the story of intellectuals, the major two being one with the Death Note ridding the world of people he considers evil to become a god and achieve order/peace, while the other opposes Kira (the former) and is out to execute him for his crimes. It's more of a mind-game series that questions if achieving peace is worth killing off evildoers, and more specifically so, WHO gets to decide what is wrong? A major point will be the viewers determining if Kira was justice or evil, both in principle and in Light's application of it (did the Death Note corrupt him too much?)... it's a bit more upbeat than Monster, and has a quicker pace IMO. It's also a shorter series in comparison.
Monster instead talks about an ingenious Japanese surgeon who is put into a morally conflicting career path. If he does not follow the director's orders, which put a value on human life (such as a wealthy scholar being put above a poor child), then he risks losing everything. Tenma, in his good nature, cannot comply with the director's orders and acts against his wishes at a crucial moment. But this decision does a lot more than destroy Tenma's life.. it accidentally "revives" the Monster of the series. After witnessing the cruel and villainous nature of this Monster, Tenma takes it upon himself to do what he never wanted to do: kill another human. This story strives off the main antagonist (the Monster), who is frightening in many aspects. His charisma and intelligence are above and beyond the norm, and his motives and intentions are unknown even to the viewers (even after finishing the series it's debatable).