The benefits of Travel granted by EU memberships are many and varied. I wouldn't want to claim that all these would disappear for us the moment the UK left the EU, but certainly none could be relied upon to stay. These rights have been negotiated within the EU. My suspicion is the Roaming Charges rule, an advantage for UK travellers only recently agreed on, would be one of the first to go.
Travellers accross Europe have rights laid down in EU legislation.
The full detais of other EU travel rights can be found here
Until recently, mobile phone companies could and did make extra charges on their customers when they used their mobile phones on their travels. They are no longer allowed to do this within the EU. Just another of those laws that we in the UK have just had to accept - Darn!
Well, maybe, but we've got them with the EU. These are the sorts of laws foistered on the poor UK population by the, ahem, democratically elected and accountable EU. I suppose, with a lot of hard negotiating, similar deals could be hammered out. I have my doubts though - What are the people of Belgium, Spain, France, Italy and so on going to say when they find UK citizens are getting rights without paying into EU funds. Unless of course, we are paying in (under a Norway style model). In which case, all we will have given up is our say in the laws.