You do have two web services. If you wanted to implement what I described today, you'd have to periodically poll Twitter's API or RSS feed for data. With web hooks, there's no more polling. Twitter calls your web service when there's new data.
Actually, now that I explain it that way, it doesn't seem that revolutionary. It just makes building neat hacks a tiny bit easier.
It sounds like this magical "push" thing already exists. It is called "get" and "post".
For example, my website is a stupid news aggregator (http://www.gibsonandlily.com). If a story gets more than ~30 points (which is really rare since there are only a few people that post there), it calls up tinyurl and says:
"Hey...tinurl, if you wouldn't mind, could you please give me a tinyurl of this link so that I can post it to twitter?"
Tinurl responds
"Hey, I'M TOO BUY POSTING TO TECHCRUNCH, GO AWAY!"
To which my server says
"But c'mon, tinurl...pleaaaseeee????"
Tinurl:
"Fine, here, no go AWAY! Also, aren't I frigging revolutionary?!"
Then it calls up twitter and says
"Hey, uhm...brah, like zomfg wutcha doin? Check out this uber link!"
and twtitter posts it.
This isn't some revolutionary "push" feature, this is just standard cgi action.
It sounds like this is the same thing, just in reverse...which really isn't that "revolutionary", more like "duh".
What you're describing is a pull, so yes, a push would be that in reverse. Polling is inconvenient. Web hooks aren't "revolutionary", but they're still cool.
Actually, now that I explain it that way, it doesn't seem that revolutionary. It just makes building neat hacks a tiny bit easier.