I am going to be teaching some people how to play D&D in a few weeks and they want to try 4e instead of trying to find some 3e books etc. Now my question is... I have the 4e DM's Guide but I was wondering if it would be better to pick up essentials vs. the other two core rules for 4e.
To get the same range of options for yourself and your players as you would getting the PH & MM, you'd want to pick up HotFL, HotFK, and Monster Vault. So you're looking at comparable cost, either way. Actually, the best deal would probably be PH and Monster Vault, you spend 55 instead of 60 or 70, your players get good class options and you get kick-ass monsters.
Is it easier to learn and teach from Essentials or about the same using the core books? What are the differences? What are the options pros and cons?
It depends on what you're trying to learn & teach. If you're trying to teach completely new players all the ins & outs of 4e, so they can eventually use the full spectrum of material out for it, starting with the Core books would give them a better persepective on how the system as a whole works. If you want to make it as easy as possible to hit the ground running, Essentials gives a simpler first experience, but has a less internally consistent scheme for class design and advancement. A player who starts with an Essentials Warpriest or Mage won't have much trouble understanding how to play a 4e character of any class. One that starts with a Slayer, Knight or Theif will have to re-learn how classes work to handle other 4e classes. Going the other way isn't even much of a problem: if you start with a 4e Fighter, you'll be able to handle a Knight or Slayer, no problem... even if you may find it a tad boring.
So, for a casual game that you don't expect to last too long: Essentials.
For something you hope grows into a long-running campaign and a serious hobby for those you're introducing: Core 4e.
Finally, if the players you're introducing are not genuinely new, but rather just haven't played D&D in a long time - since the 90s or even 80s - then Essentials has another advantage, in that it's wrapped in a coating of nostalgia. It's little things, like the cover design of the Red Box, the names of some classes and features, and the way the classes vary in complexity and situational effectiveness. Old-school players coming back to D&D expect the fighter to be big and dumb, and the magic-user to start out weak and become godlike. 4e is paradigm-shattering for them. Essentials lets them down easy.