This is indeed both mystifying and slighty annoying, seeing as Chrome's sync feature offers just that.
I would have to test that (just an idea that popped into my head): You could sync your complete Firefox profile folder using a cloud-based folder syncing tool such as Dropbox by placing your Firefox profile folder on both PCs insided the folder synced by Dropbox - you would probably have to rename your profile folders to have the same GUID, but it could totally work.
On googling that mid-way into my trail of thought, I happend upon an article on Lifehacker that pulls off something like that. It's bit old-ish, so it doesn't include either Dropbox or Firefox sync. I'll try to work out how to do that some time later today in more detail (now that you've brought it up, I'm really curious about getting that to work), but that's the general idea, and I think it should work.