CPU – Identify Intel Generation on Mac

They are different processors.

The Core Duo was released earlier in 2006. The Core 2 Duo was released mid 2006.

For specific dates, and more info on the processors, this Wiki page is a good reference: https://en.wikipedia.org/wiki/Intel_Core#Core_Duo

Firstly: putting your serial number up for all to see may not be the best idea.

Your Mac Pro is an original Mac Pro. This means that it is not compatible with El Cap by default (I'll explain more later.) So yes, there's an error on your part.

Your MacBook Air is also (sadly) likely about as powerful as that Mac Pro. If you upgrade it, it's possible that you may find a different story, but regardless, that Mac is reaching end-of-life. Depending on your use case, it may be fine to upgrade it, but there's not a lot you can do. Instead, I'd recommend that you save up for either a more powerful PC or Mac.

With that said, you can make your Mac Pro work on El Cap if you really want to. It's a little tricky, but possible.

Your Mac is incompatible because it has a 32-bit EFI firmware and an outdated GPU. Both of these are fixable issues. I would recommend reading this article, but I'll summarize here. By using a different bootloader, you can circumvent issues. Follow the instructions on said page. Your GPU is also unsupported. This means you'll need a new GPU. This is more complicated then you think however. Some cards will require being "flashed" as though they're Mac GPUs, while others can do most (but not all) things without it (and can be flashed for more compatibility.) Again, follow the guide in the article.

I hope this was helpful. Again, this machine–unless it's a fun project–probably isn't worth your time or money. But hey, if you want, go ahead.