what comes up as an open wifi network isn't. On a laptop, connecting to the network launches a web browser, which asks you to either log in to your account or buy network time.

I've found on my smartphone that connecting doesn't launch the browser, so the status is connected but you can't download anything.

I've had a similar problem with an ipod touch, more than once, which was cured by resetting it, not the factory reset, just the one that causes the Apple logo to appear. Have you tried that?

'other wi-fi', what kind of networks are you referring to? Friends' houses, or shops and cafes?

Have you tried changing the SSID and password of your router so that you need to re-set up your ipod to log into it? Even briefly setting it as an open network?
If you then can't log into your router you'll know that it's not a "settings" problem because the settings will be the same as before even though the router effectively has a new identity. Obviously that won't solve the problem but it will help narrow down the cause and could provide a method of knowing when you've fixed it while you're still at home.
Note: It's always best to change wireless settings from a wired PC just in case something goes wrong.