How they are different
Everyone, including the documentation, tends to refer back to how much they are the same, but not actually reference any differences. Well, they are, in fact, different.
var bigApp = express();
var miniApp = express.Router();
listen()
The most obviously difference is that the bigApp will give listen, which just a rather confusing way to do what would otherwise be simple and obvious the node http or https module:
var server = require('http').createServer(bigApp);
server.listen(8080, function () {
  console.info(server.address());  
});
I consider this an anti-pattern because it abstracts and obscures away something that wasn't complicated or difficult in the first place, and then makes it difficult for people to use websockets and other middleware that require the raw http server.
Internal State
The big difference, which is really important, is that all bigApps have separate internal state.
bigApp.enable('trust proxy');
bigApp.enabled('trust proxy');
// true
var bigApp2 = express();
bigApp2.enabled('trust proxy');
// false
bigApp.use('/bunnies', bigApp2);
// WRONG! '/bunnies' will NOT trust proxies
A miniApp passed to a bigApp, however, will be operated by the bigApp in such a way that its internal state and thisness will be preserved and those routes will behave accordingly.
bigApp.enable('trust proxy');
bigApp.enabled('trust proxy');
// true
var miniApp = express.Router();
bigApp.use('/bunnies', miniApp);
// CORRECT! All state and such are preserved
This can be a big deal because express does a lot of (sometimes trixy) things to the http.ServerRequest and httpServerResponse object - such as modifying (or hijacking) req.url and req.originalUrl and various other properties you've been using without realizing - and you probably don't want that duplicated and separated.
Smaller API
There is a smaller, more well-defined number of functions a Router can use:
.use(mount, fn) 
.all(mount, fn) 
.options(mount, fn) 
.head(mount, fn) 
.get(mount, fn) 
.post(mount, fn) 
.patch(mount, fn) 
.put(mount, fn) 
.delete(mount, fn) 
.route(mount).XXXX 
.param(name, cb).XXXX 
There are a few other convenience methods as well, such as basic(), but you won't find set() or enable() or other methods that change the larger app state.