I have been playing Dungeons & Dragons since the first edition of Advanced Dungeons & Dragons. That first edition came in 3 hardcover books, the Player's Handbook, the Dungeon Master's Guide, and the Monster Manual. And these 3 core books have been how AD&D (renamed D&D since the third edition) was released in every edition up to and including the 4th edition of 2008. That changed in 2010 where a new line of products was released called D&D Essentials. While D&D Essentials are still 4th edition rules, the books are now smaller sized paperbacks, there are more of them, and they aren't called Player's Handbook, Dungeon Master's Guide, and Monster Manual any more.
The easiest case is the Monster Manual, which has been replaced by the Monster Vault. While the Monster Vault book is a small paperback, what you buy is actually a big box which contains that paperback and lots cardboard sheets with tokens for all of these monsters. Plus there is an adventure and poster maps. As Amazon sells the box for under $20, that is quite good value for money. The only downside is that there are less monsters in the Monster Vault than in the Monster Manual.
The Dungeon Master's Guide also comes in a box with a thin paperback book on how to be a DM, a DM screen, an adventure in two parts, a monster booklet, monster token sheets, and battle maps. As you can see, there is already some overlap with the Monster Vault. Again the idea is more to have the whole game in a box than to offer exactly what was contained in the Dungeon Master's Guides before.
An interesting addition to all that is the Rules Compendium, a 320 pages paperback which contains all the 4th edition rules as a reference. As since 2008 the 4E rules have had errata and addendum, this comes in handy. And I actually like the small paperback format better for a rules reference book.
All these D&D Essential products mentioned up to now are completely compatible with the 2008 Player's Handbook, Dungeon Master's Guide, and Monster Manual. There has been some rebalancing of monster stats and things like difficulty checks, but essentially (pun intended) this is still 4th edition D&D.
Where D&D Essentials differs a lot from the 2008 4th edition is in what used to be the Player's Handbook. This now comes in two small but thick paperbacks: Heroes of the Fallen Lands, and Heroes of the Forgotten Kingdoms. Each book has only 4 character classes, and different races. Thus you would need to buy both to get the 8 classes of the Player's Handbook, and you'd pay more for that, and get a lot of rules explanations twice.
Even worse, the characters you can produce with the D&D Essentials books for players differ significantly from the characters you could produce with the Player's Handbook. Basically the D&D Essentials books offer you a lot less choice. Take for example a fighter / knight: In the PH a fighter would have daily powers, which got removed in D&D Essentials. A PH figher would get the choice of 1 out of 4 possible encounter powers at level 1, and another choice of 1 out of 6 possible encounter powers at level 3. In D&D Essentials he gets Power Strike at level 1 instead, no choice, and a second Power Strike per Encounter at level 3, again no choice. Only with the utility powers the choice is similar. But even then the powers are not exactly the same. The D&D Essentials characters are not weaker, and could theoretically play in the same campaign as the original ones. But you try to explain your players then why some of them have choices when they level up, and others don't.
Fortunately for my particular campaign the question whether players can use D&D Essentials characters doesn't pose itself. The D&D Essentials haven't been translated into French, and some of my players don't speak English. So my campaign is based on the French versions of the Player's Handbook and Dungeon Master's Guide. And even those had limited availability, apparently D&D isn't doing so great in France.
Overall I love the Monster Vault and the Rules Compendium, and I also ordered the Dungeon Master's Kit. I'll stay away from the "Heroes" books of the D&D Essentials, as I don't like the simplified character creation. I recognize the efforts to make the game "more accessible" (where have I heard that before?), but I think WoTC went too far here. Choosing powers on character creation and leveling is fun, and reducing that fun and the choices is not a good idea.