First, defining what "romantic" means would be a chore in itself.
Assuming we can all sort of vaguely, generally get the gist of it, my guess would be that most of the generalized behaviors we see are culturally enforced; that is, men are implicitly guided to be less romantic because it is perceived as less manly. Absent cultural influences, the distinctions might be very small.
On the other hand, I'm not convinced these distinctions are completely fabricated by culture, either. In almost all areas I've seen studies done, differences between male and female sexually have been observed even when cultural variables are controlled for as best possible. Further, as we all know, men and women have different sex hormones running through their bodies, so it shouldn't be surprising if different sexual behaviors are generally exhibited between the two sexes.
So to stop pontificating and sum up my point: the "romantic" differences between the sexes are mostly -- but probably not entirely -- a construct of our society. There are some noted differences between male and female sexuality and I'm generally wary of assumed absolute equivalence between the genders in this regard.