I agree with Tucker; it looks like a lamium. Is it really a vine? Lamium usually grows as a groundcover. If it is a vine, maybe it's not lamium.

Thanks, zubababy!

The 'prickles' aren't very big -- they aren't big thorns like you would see on a rose or a blackberry stem. One description called the prickles 'stiff hairs'. The web site below calls them 'hooked hairs', which sounds close to your description that called them velcro. Apparently the vines use these hairs to cling and climb on things. I was amazed to find that people plant these things in their gardens as ornamentals, and the web site below calls them "desirable gardening and commercial species"! Hope you can get rid of yours. --BC -- Here is a link that might be useful: Humulus japonicus vines are covered with hooked hairs

Maybe it is one of the Campsis species? I have a few, and some of them have leaves that match your description. Here is a link that might be useful: Some Campsis species listed here.