What are the advantages of the mac mini's hdmi output? I have a late 2008 MacBook so I know that if I want to connect my laptop to a tv I need two cables: one for video and one for audio (while in newer models this issue is bypassable via appropriate adapters). So using the hdmi port of the mini would make my life easier. 

Are there any other advantages?

For example when I first connected my laptop I needed to do some fiddling around with the displays settings to achieve a decent output: would that be necessary with the mini as well?

I've been using laptops for so long I forgot what it's like not having a display already embedded in your computer!
If you keep your machine plugged into one display, there won't be much of any 'fiddling'. And even if you switch, 10.7 actually is supposed to help with switching displays (as in, if you go from a LCD monitor to a TV). 

Another advantage is that the machine was just updated, so depending on how much you are willing to pay, you are going to get a more powerful machine than what you have now. 

As you said, you will also have HDMI out, which will contain audio as well. 

Finally, you will need to purchase a display, mouse, and keyboard if you don't already have one.