Some 30 years ago I used to play Dungeons and Dragons: red 'Basic', blue 'Expert', black 'Master' edition box sets, etc. I still have them and was recently playing with my son, who loved them. I want to get him his own sets, and thought the D&D essentials Monster Vault looked good. My question is, would this set be compatible with my old sets in any ways? Thank you for your help!
Current editions of D&D are not at all compatible with Basic D&D. The new editions are based on the old editions in a very tangential way, each one deliberately transforming the game compared to the edition that went before, but there have been so many superficial and fundamental changes to the rules that they are now completely incompatible. Any conversion between Basic/Expert/Master D&D and D&D 4th Edition or even 3rd Edition would be more of a re-imagining than conversion, and would take as much work as converting to a completely foreign system.
However, your actual goal is getting your son his own copies of what you're already playing. In that case, you're in luck! Lots of people never stopped playing those editions. "Retroclones" of them are readily available, often for free (if you're OK with digital PDF rules), and frequently in print at reasonable prices. These were created based on the principle that you can copyright a game's words but you can't copyright the rules, so people just rewrote the games. There is a thriving community built around these retroclones.
There is a long list of retroclones out there (the list of retroclones at Wikipedia is not exhaustive at all). For the closest one to Basic/Expert/Master (also known as "BECMI D&D"), the leading choice is Dark Dungeons, which is available as a free download and in print.
However, if you're OK with PDF-only options, you can actually still buy the Rules Cyclopedia (which is essentially the BECMI books all edited together into one volume) as a PDF from the current publisher (who bought TSR back in the late 90s).
No, it would not be, at least not easily.
But it's all a matter of degree. Older versions of D&D, like BECMI (red box... immortals), are a little more cross-compatible with earlier and later versions from back in the day but it was still pretty different from say AD&D 1e. Each revision has gotten more different - 2e, 3e, now 4e, and 4e is probably the most different of them all. The Monster Vault is 4e and other than "mining it for ideas," it's not compatible with the other versions.
You can get him his own BECMI set, at least in PDF, because Wizards is selling their older editions again at http://www.dndclassics.com/. Or, you can get him started on a newer version - 4e, or actually it might be better to just start with the free D&D Next (5e) playtest - it's actually more back compatible than 4e is and inherits some of the Basic/2e approach.
There are all the retroclones, but I'm not sure why you'd get one of those when the originals are available again. See Overview of D&D retro-clones for some options there.
Though the new D&D market leader isn't even D&D any more, it's a variant called Pathfinder and is preferred by many. It has a beginner box product and a large well supported game line. You can check that out too.
But if you have a preference for hard copies and would like to get as close to the original intent of old-school Dungeons and Dragons without a box set or retro "clones", you can actually buy hard-back (non-pdf, genuine dead tree) reprints of the both the original Advanced Dungeons and Dragons 1st Edition and 2nd Edition rule books from Wizards of the Coast.
Personally, being my introduction to RPGs in general, I would highly recommend this avenue. They are much more likely to be in line with what your are describing rule-wise and in game tone.
That said, one advantage of purchasing the PDF-only Dungeons and Dragons Rules Cyclopedia as mentioned by SevenSidedDie is for it's "extra features" including:
Rules to convert D&D games and characters into AD&D 2nd edition game statistics and back again
Thus you and your son could have the best of both worlds.
Thanks to SevenSidedDie for pointing out both editions were available as reprints, and that indeed, Advanced 1st Edition is even closer (more compatible) a rule set to Basic D&D than Advanced 2nd Edition.
The simple answer is that there are three board families of games branded with D&D.
Classic D&D which includes OD&D, Mentzer's BECMI D&D, AD&D 1st, AD&D 2nd and current playtest version of D&D 5e/Next.
D20 D&D which includes D&D 3.0, D&D 3.5, and Pathfinder
D&D 4e, D&D 4e Essentials
Understand there are differences but they are a matter of inches and within each family support material generally just work as is.
D20 D&D and Classic D&D can use each other adventures and settings with a little work.
D&D 4e settings and adventures can be used with earlier editions with work. However the balance of Monsters is very different but on the plus side 4e adventure often come with battlemaps and other useful props. The reverse unfortunately is not often true due to amount of time it takes resolve a combat encounter. The dynamics of 4e Powers and Healing Surges means that combat often takes longer to resolve than earlier editions.