Windows 7 not detecting external hard drive but Ubuntu is detecting. Why

The times I have seen this it was a disk fault, not necessarily hardware. On the machine that assigns it a drive letter, one cheap and easy technique is to just run chkdsk /f on it.

I agree with the previous recommendation to disconnect (only while actually installing Ubuntu) your internal hard drive, to be certain that your internal hard drive boot sector is not changed.

If you select the automatic/easy install method, then the default action by Ubuntu is to change your internal hard drive boot sector. You can avoid that by selecting the Advanced option when installing. Then you can tell it where to install the boot loader. The boot loader must be installed, even if you will be using your bios booting option to choose to boot from the external drive. In your case you would want to install the boot loader onto the MBR of the EXTERNAL hard drive. Note that the boot loader should not be installed into any partition. Also, when installing, I recommend disconnecting from the internet and choosing not to install updates during the install. Then do the updating only after you confirm that the new install actually boots OK. This avoids the possibility that you spend an extra hour for the getting updates, only to maybe find out that it won't boot for some reason. I'm still learning too.