No internet when booting from Windows to OS X I installed Windows with Bootcamp and have an Airport Express. Every time I boot from Windows to OS X it connects to the wireless network but not to the internet. Other devices still keep their connection to the internet.

So, the only thing that makes this work is to turn Macbook's Wi-Fi off, turn Airport off (!) and turn on everything again.

The same problem also happens when I boot from OS X to Windows, but in this case is easier, since I only have to turn off and on the Wi-Fi.

The problem doesn't happen with my Linksys router.

I'm using Moutain Lion 10.8.2, Windows 7 x64 and I think I have the last Airport firmware version: 7.6.2. I'm using DHCP.

Do I have to click on the Bootcamp tool and chose Restart in OS X... option or can I simply restart the computer using the Start button?
It sounds to me more like your Airport Express is acting up (since you mentioned that the Mac works fine with your older LinkSys router.) Check for a firmware update via Airport Utility. 
Additionally, you can force Windows to release the DHCP lease on its IP address by going in to the Windows command prompt and typing 

C:\Users\myusername&gt; ipconfig /release


Just do that before you restart into Mac OS X. I agree with @Buscar that you should try to reinstall the BootCamp drivers (and check Windows Update for any updated drivers.)