The live-ness of tiles is overstated, in my opinion, but the concept of tiles is wonderful. Here's an example: you're going on a trip so you pin restaurants that look interesting from Urbanspoon, a map to your meeting location from Maps, and some sites you want to see from their web pages. It turns your home screen into a todo list, which is interesting to me.
Ah. From that description it sounds like there's not a real difference in what widgets allow apps to do, but there is in how easy it is to add and utilize widgets (with Windows being at one extreme, where all or most app icons are widgets)?
Yeah, a Live Tile is a widget, but it's a well integrated widget. It's an application shortcut that displays information while still looking like an application shortcut. Widgets are completely separate, visually. Windows Live Tiles don't need to be specifically added, if you add an app shortcut to your start screen it's automatically also a widget that looks visually consistent with the rest of the OS.
Your assessment is not too far off. Live Tiles is a more elegant way of handling widgets.
Sort of, I guess the biggest difference is in how you add them. Adding widgets is an intentional thing; you decide you want a weather widget on your screen. A tile is something you discover and decide to keep for later. So you find a restaurant you like, you pin it. You don't say "I want to pin something, let me look through a list of pinnable things and pick one".
That's one use for Tiles, but not the one I'm referring to. Pinning a map is great, but pinning a weather app makes it continuously update the local weather and forecast right on the app shortcut. Pinning a stock app makes your stock quotes show up right there on the app shortcut. There's many use cases for pinning a Live Tile, and the best part is they're all done using the same mechanism.
To do something like this on Android, you'd need an API for an app to install a widget. I don't know if that's possible outside of the standard appstore download/install process.
taude already explained it but I will add my two cents. With widgets the look depends on the developer, which I found annoying when I had an android since I had to go through several screens just because some widgets where really big.
Live tiles are "free", they are the same icon that you would use to start the app and the look and feel is more uniform and rearrange between them in a more convenient way.
Also you can pin a lot of app specific items. For instance, from the people hub I can pin a tile of the whole app and/or pin specific people, and when you do that the tile itself seems like a different app. In my wife's tile I can see whatever she posted on Facebook or twitter, and also it serves as a shortcut to publish something on the wall, or to call or text her, it's just very convenient!