Without seeing the plant, its difficult to tell but the basic leaf shape is Ivy shaped but this "thinner vine that grabs onto anything" suggests to me that it may not be an Ivy at all.
The thinner vine may well be a tendril, which Ivies dont have, just small anchor roots for airial support.
We have something that grows wild here in the UK called White Bryony which sounds similar but I would'nt put money on it.
Would it be possible to post a photo, for a better chance of ID. Good Luck Tbird+