Both are even made by the same manufacturer: HTC.

In short: HTC Desire = Nexus One - Dual Mic + Sense UI + Optical Trackball.
HTC dumps dual mic (for noise cancelling) because it would drive the cost down compared with Nexus One, a good trade off in my opinion. You could easily get bluetooth headset with noise cancelling if you really want that feature.
Both are using Android, but HTC has added an extra user interface, the Sense UI, which may be why you thought they ran on different OS's. Sense UI is basically a heavily customized interface made by HTC, and a very good one at that, on par with the iPhone interface in many people's opinion.
Since Sense UI is made by HTC, it would not be available on other Android phones like Motorola Droid, since lots of the hardware would be incompatible with the UI. It would be like installing cellphone OS to your GPS gadget: both are capable, but have different hardware in it.
I would pick HTC Desire rather than Nexus One. It's cheaper and better compared to Nexus One. But if you live in the states, you might want to factor in the availability of HTC Desire in deciding which phone suits you best. This is because the current model that would be launched in April 2010 don't support US 3G network, so you would be stuck in 2G. As with HTC HD2, the US version would probably launch latter this year.