The definitions seem to be pretty distinct:
Romance (n): A quality or feeling of mystery, excitement, and remoteness from everyday life.
Adventure (n): An unusual and exciting or daring experience; A reckless or potentially hazardous action or enterprise.
Romance deals with something that has the attraction of being pleasantly unusual and mysterious, something very different from the kind of thing you might do every day. Often this involves some measure of risk -- not necessarily physical risk, but rather the courage to try something entirely new, to "wander off the beaten path."
Adventure, on the other hand, often has some qualities of romance, but combined with the element of physical risk. Most adventures contain some chance of injury or death.
So, for Americans, traveling around Europe on a bus tour might be considered romantic, because we get to see many famous places and have unusual experiences ... but not necessarily adventurous, because there's little danger. It's considerably more adventurous to backpack around Europe by train, staying in youth hostels and cheap hotels, because then there is much less control over what might happen.
On the other hand, enlisting in the military is considered adventurous but not particularly romantic (depending on your point of view) because most imagine the experience will be unusual but not pleasant, and there may be considerable physical risk.