A new graphics card I am looking at has a mini HDMI output and I wondered if anyone has an HDMI link to their monitor and whether it give better picture etc. don't want to buy a cable if it won't improve it somehow.

You can get an HDMI cable for a couple of quid. The advantages are that the plugs have a smaller profile, so easier to fit at the back of equipment, it carries audio if your monitor has speakers, and it facilitates HDCP protected playback from commercial Blu-Ray etc. There won't be any difference in image quality (unless you already have a dodgy DVI cable/socket), but HDMI does have more bandwith available.

All I have been using is HDMI. Have a DVI to HDMI adapter on my Zotac GTX 580 AMP2 3gb. Card has
mini HDMI and they sent along a converter. I prefer the more solid mount of the DVI to HDMI
adapter of the other. I am using a Insignia 42" LED 1080P 60hz monitor. Looks fantastic using
HDMI.

Using DVI to DVI on my PC -> Monitor, if I use the mini HDMI I loose the audio out from the PC therefore can't use my headphones and then the familly complain that my games are disturbing them Sigh. not seen any diffrence in pic quality between DVI and HDMI to be honest, and HDCP works over DVI too as long as everything in the chain is compliant.
I have an Asus GTX580 card, if I connect the supplied DVI to VGA to connect to my TV I'm limited in resolution and the TV won't display 1920x1080 but has no problem with either DVI or HDMI and dispalys 1080p fine, TV is a Pioneer 5090 50", PC monitor is a Samsung 24" LED