Looks like Clematis armandii. You don't have to do anything with the snowdrops, just leave them be.

Don't know the clematis, but leave you snowdrops to die back now, leaves as well. Give them a feed and that will help the bulb for next year.
Someone will know this clematis.

Sorry, Bamboo, we crossed again!

Yes I'd go with armandii too, its a lovely clematis.

I just leave the snowdrops to die back. in about 6 weeks the leaves will die off and you can remve them. A feed is a good idea too as Ojb suggests.

thanks for all your help!!