I've played mostly 2nd Ed. and some 3/3.5. I have a lot invested in those materials. I am intrigued by 4E an Essentials, but am looking for the best way to approach the new editions. Keep in mind I am introducing my children of varied ages to D&D as well. The older ones have limited experience with the game, while the younger ones will be new to it. I am a little confused about the differences between 4E and Essentials. Should I start with Essentials and then move to the core rule books, or just start with the core rules?
The two have the same rule base and are completely compatible.
However, if you are just starting, pick up either or both of the Essentials books Heroes of the Fallen Lands and Heroes of the Forgotten Kingdoms (depending on which classes/races you are interested in. As a DM, pick up the Dungeon Master's Toolkit, the Monster Vault can wait until you have a few games under your belt and are sure you will be continuing. The Rules Compendium isn't needed as all the rules are in each of the Hero's books.
The main problem with the old trilogy (Player's Handbook, Dungeon Master's Guide and the Monster Manual) is that a considerable amount of errata has been issued since those books were released. I'd recommend skipping them until you have the rules well in hand and can grab those books for any options you want.
This is just to answer the titular question: What is the real difference between D&D 4th Edition and Essentials?
Essentials is a beginners version with fewer races and classes and most of the options removed. So instead of a core 4e Fighter (which might be any one of several builds, and gets a choice of one of several options at each level), the Essentials Knight gets a predetermined option or might have a choice between a smaller set of abilities.
Essentials versions of the classes also have a simpler mechanic by which they (mostly) use their basic attacks and stances instead of the choices a core PC gets.
The real difference is this: Essentials is simpler, but has fewer options. It's a beginners thing— they are the same game.
4e Essentials is fully compatible to non-Essentials 4e.
Since you are introducing your children, (I did the same with my 3 sons about 10 years ago), I would recommend getting the D&D Starter 'Red' Box. It takes you through everything a newbie would need to grasp the game. It has a structured learning system that has you actually playing the game while you are learning. It is excellent in that regard. You will find that you can go through it and exhaust what is in it in about 2 weekends.
After that move them to Heroes of the Fallen Lands. It follows nicely and expands on the four basic character types that are in the Red Box. Getting a Dungeon Masters Toolkit at this time will expand the game out to about 3rd level.
Once you are ready to take them beyond that you can add the Monster Vault and Heroes of the Forgotten Kingdoms for more character types. The rules compendium puts everything into a single place that is well organized and easy to handle.
The most important thing for you to know is that if you pick up the Red Box, or the Rules Compendium and one of the "Heroes of" books, you have the 4E rules and will be learning to play 4E. In addition to having the balancing errata after 2 years on the market, the Rules Compendium is a much better read. After playing 4E from the initial release, the RC gave my group a lot of clarity in different areas of the rules. I would recommend to anyone new to the game to start with the Essentials products.
While some might describe the essentials classes as 'simple' or 'basic', in my opinion a better word is 'cleaner'. They're also more distinctive - between the two Heroes books, you'll get a pretty good selection of classes that really play differently from one another. Yes, there's not as much complexity as the 'Core' Players' Handbooks - but you can easily add that on later.
To be honest, I prefer to play Essentials builds in my current campaigns. I find combat is a little more streamlined with fewer bookkeeping powers/buffs. And they play just fine alongside any 'non-essentials' builds.
As an answer to the question in the title, the Essentials line is full of products that any store that wants D&D can carry successfully, while the D&D 4E line is best suited for hobby and game stores. As a retailer, you can stock Essentials products and be assured of regular turns. Books in the 4E line other than the core three are designed for niches of a small market, and might simply gather dust on the shelves at Target or Wal•Mart.
The rules are compatible and many of the class options can be mixed between versions.
The real difference is that Essentials limits itself to the traditional spread of classes and races, including some tweaks to how powers are gained and used that preserves some of the play feel of the pre-4e versions of the core classes. By contrast, the full 4e line allows players to create such off-genre things as Warforged Runepriests, Dwarven Invoker/Swordmages, and Shardmind Battleminds. Along with those less traditional possibilities though, the full line gives players a much wider choice of races and classes from which to build their character, which adds a "meta" character-building/advancing, strategic component to the game which is to many players' taste.
Depending on taste, having a full palette to create characters with might be a pro for your group or it might be an unmitigated DM's nightmare. Choosing to start with (or stick with only) Essentials over 4e or vice versa gives you the chance to twist that dial to your preference.
You probably want to just use Essentials.
Though the rules from original 4e to Essentials are "compatible," Essentials incorporates a bunch of errata and has basically rethought designs for the core classes. And if you get the four core Essentials books there's very little you are missing from the original core 3 4e books, there's a lot of literal and a while bunch of conceptual overlap. Mainly you'd get them for obsessive completeness, because there's one sweet power or monster you want from them, or to be compatible with a group that has a lot of them.
Actually all the classes are in essentials and all of the races. However, no there are not as many items and at the moment no multiclassing rules/hybrids and weak ritual rules. Plus with the release of the Sword and Spell compendium for essentials then all those missing rules/other builds will all be available.