The most obsolete books are the monster ones (MM1, MM2, Draconomicons), though the older monster books can be updated to the current state of the art with some mechanical adjustments.
The older players books are not made obsolete by the essentials players books, but they also don't mesh that well with the essentials books. Players playing a character from one source will not find all that much use for the other source.
The advice portions of the DMG books (which accounts for a large segment of the books) overlaps somewhat with the DM Kit book, but not enough to obsolete them.
The three Player's Handbook books (and the very related power source books - Arcane Power, Primal Power, Divine Power, Psionic Power, Martial Power and Martial Power 2) present character classes that are built somewhat differently than the character classes in the two essentials players books.
Classes from the earlier sets of player books can be played alongside the classes from the essentials books, but players of some classes (especially the martial ones) built using one group of books will find that the options presented in the other group of books for their "class" are of little use.
Player's Handbook 3 and to a lesser extent Player's Handbook 2 present options for which there are no essentials line alternatives. None of the psionic classes from PH3 are found in essentials, nor are the Bard, Avenger and others from PH2, though the essentials Slayer Fighter plays a lot like a simplified Barbarian from PH2.
Races received some updates in the essentials player books. For example, humans from PH1 get an extra at will attack, humans from essentials get a racial power instead. In terms of official WotC ruling, a human can pick either option.
Heroes of Shadow, the most recent players option book presents new classes and races, and also presents options, some of which are applicable to the classes as outlined in the essentials player books and others more applicable to classes as outlines in the original player books.
The essentials player books also introduced the notion of magic item rarity, but otherwise made no substantial changes to the way magic items work, so magic items from any of the players books (original or essentials), as well as from the Adventurer's Vaults should be usable.
The Dungeon Master Guides are primarily about advice rather than rules, though a substantial portion of the rules presented in the first DMG - treasure awards - is replaced in the essentials Dungeon Master's Kit, and much of the advice to dungeon masters presented in DM Kit book is from the DMG1 (which I believe cribbed some content from earlier editions DMGs)
Monster Manual 3 introduced higher damage numbers for monsters, as well as a new design philosophy for monsters especially elites and solo, as well as a new layout for monsters, different from what was found in Monster Manual and Monster Manual 2. The Dark Sun Creature Catalog, Demonomicon, Monster Vault (from the essentials line) and all subsequent products all follow the MM3 guidelines.
MM1, MM2 as well as both dragon books use the older layouts and design philosophy.
Other than damage, the most obvious changes are the reduction in the frequency of monster powers that "kill the fun" for players (dazed, stun, dominate) and a significant increase in the ability of elite and especially solo monsters to shake off or work around such effects.
Finally, if you're using the tokens that came with the Monster Vault a consideration may be that only the essentials products, the starter sets, and the more recent boxed-ish products: Gloomwrought, and Monster Vault: Threats to the Nentir Vale also come with tokens.
There are some differences I've seen in the various token sets - The older, blue starter set, as well as the red box put different monsters on each side of the token, while the DM Kit and the Monster Vault put the monster outlined in red, to indicate bloodied. Gloomwrought, and Monster Vault: Threats to the Nentir Vale continue this trend, but also write the monster's name on the bloodied side.