Well, would it be a surprise if people told you to have a look and see which you prefer? That seems too simple though so I'll tell you what I think, since you asked. And then the following posts will tell you that I'm totally full of beans and should be ignored. They will then proceed to tell you the opposite of what I've said and you'll end up having to make up your own mind anyway.
Anyway. Here are my thoughts on random things, some of which may be relevant to you.
The LCD versus plasma debate rages on unabated. All the while, plasma is slowly disappearing. Pioneer at the top end and Vizio at the low end have recently abandoned it. For this reason, you can find some good deals on plasma if that is what you prefer. I never advise buying cheap unless you have to though. You get what you pay for. It's not the screen that makes Pioneer TVs really good; it's the electronics in them. Cheap plasmas just don't have it and that shows up in picture quality.
I appreciate the richer colours and deeper blacks of plasma, but I can't stand those honking big pixel grids I see when the picture is white or near white. It's like looking at a picture through a screen door. LCDs have smaller pixels so it's not as bad on that point.
The advantages of plasma are eroded by LCD with each generation, but the advantages of LCD are not eroded by plasma. Hence the slow death of plasma. This leads to the point of your original post, i.e 240 hz versus 120 hz processing.
Two new features in the latest generation of LCDs are the LED backlighting and the 240 hz processing. But you have to pay for them. The corollary is that you can get the older generation sets for less money. So, is it worth the extra money?
Generally speaking, anything with speed is good. A lower response time and faster processing reduces artifacts in pictures that move fast, like action movies or sports. Do you see artifacts in 120 hz sets? If so, you will notice an improvement with 240 hz. If this is not a problem for you, the 240 hz probably isn't worth paying a premium for at this point. So you have to look for yourself to decide this.
The other new feature, the LED backlighting is, in my view, a much more significant improvement. The colour is more continuous than with fluoresecent backlighting, and richer. Dare I say it?...it's more "plasma" like. That's what I mean when I say that each generation of LCD erodes the advantages of plasma. The improvement is so significant that I think LED backlighting will quickly become the norm for LCDs. But you will pay for it until all the old stock is sold and everybody adopts it and drives the prices down.
So what to do? If I were buying a TV right now, as opposed to six months from now. I wouldn't spend top dollar on an LCD just to get 240 hz. You can get good deals on the older generation. For what I watch, I don't find the jump from 120 hz to 240 hz worth the premium. LED backlighting is what would make me lean towards the more expensive LCD set. On the other hand, if you prefer plasma rather than LCD, you can get equally good deals on plasmas right now although for different marketing reasons than the LCD case. Look for discounted Pioneers, or if they're too much, you never mentioned Panasonic. They make excellent plasmas too, at a good price. And plasmas are not only nice to look at; you can heat your room with them too! And they won't increase your electricity bill by any more than $50 or $60 a month (Well, obviously this is an exageration, it could be more or less than that).
Wait till you see laser and OLED TVs.