Hard question to answer as really, almost anything would be great depending on what they like, what prior experience they have with live theatre, etc. Have they already seen touring productions of Phantom, Chicago or Mamma Mia? Those shows have been running for a long time for a reason. Don't rule out Lion King or Wicked for either teen-they are both terrific shows that are not just for kids. Good friend of mine-a high school senior- just got back from seeing Once and loved it (I did as well). It has a "newer" feel and hasn't toured yet so they'd be cutting edge. Same of course with Book of Mormon. I've not seen spiderman so won't comment there. And I'm making the assumption that by Broadway you mean musicals. But if that isn't the case then look at a listing of plays too. The ones I'd suggest now all seem to be closing in feb/mar, so you may want to wait a bit to see what's opening. As for me-my daughter and I are thrilled to be seeing Matilda (based on the Roald Dahl book) when we go in March. But it hasn't opened yet so be aware that you might not like what the reviews have to say - we personally aren't worried and already have tickets, but t would be a lot of money to spend if it turned out to not be what you thought.

Buy discount tickets in advance at broadwaybox.com. You will have a better choice of seats and a better change of getting 4 together if that's what you want. And why waste and hour standing on line while you're here.

As for what shows - what have they seen? Are you looking for great dance? Or an extravaganza? Or ????

All would love Book of Mormon - but tickets may still be big $.

I, personally think Lion King is for little kids - but I know others disagree (I must admit I don't get Disney at all.)

First, let me correct a possible misleading statement of the previous poster, who says, "The ones I'd suggest now all seem to be closing in feb/mar." I don't know what he/she was referring to, but the only musicals that I know of that are closing before the OP's visit are The Mystery of Edwin Drood and Mary Poppins. So, actually, all the shows he/she mentions will still be playing.
There are many "appropriate" shows currently for teenagers. If you're looking bargains, then start by checking out the discount offers on broadwaybox.com and playbill.com. You're better off using that route for discounts rather than TKTS. The price is often not that much different, and you save a lot time by not having to wait in line at TKTS.
Suggestions? It's hard to make them without know more about the tastes of your teenagers. I'll make a start, however--for her: Wicked; for him: Spider Man. Discounts are available for Spider Man, but not for Wicked.

Actually - I was saying if they also wanted to look at plays (as opposed to musicals) then to look at a listing of what is playing/coming. The straight plays that I would suggest for now would be Cat on a Hot Tin Roof, Who's Afraid of Virgina Woolf, The Heiress - looks like these are closing soon. I didn't want to assume that the OP only meant to look for musicals. And I will disagree on only recommending Wicked for the daughter. my then-teenage son really enjoyed it. And I know he'd love to see Matilda (now, at the age of 23) being a Roald Dahl fan.

First all, I don't think that those three dramas would be that appealing to teenagers, especially when compared to the available musicals.
Further, I wasn't only recommending Wicked. I just think it's a good starting point for a teenage girl. (And, this comes from someone who did NOT like Wicked!)
Before I add any recommendations, I'd certainly would want to know at least a little more about the tastes of the teenagers.

Newsies, Once, Pippin, Kinky Boots are some ideas. Playbill.com has a list of what's on and discounts. Didhelikeit.com will gove you vritics reviews of things already open.