This is a good question. Romanticism was an artistic and intellectual movement that started in the 18th century as a reaction against the growing backdrop of the industrial revolution. It was a movement, as its name suggests, that embraced strong emotions of all kinds as a important source for art. This emphasis was also opposed to the rationalism of the time.
Postmodernism is a philosophic and intellectual movement that started in the 20th century and it is very well alive even today. It challenged the rationalistic framework of the modern period. More specifically it challenged the objectivity of knowledge and hence became very suspicious of any sort of meta-narrative. Postmodernism also is very aware of power structures and how they work.
There are many differences between the two movements. First, they took shape at different time periods and they possessed a challenge to two different movements in society. Second, romanticism emphasized the emotions, where as postmodernism did not have that emphasis. Third, postmodernism would make relative the emphasizes of romanticism as a construct of a certain time period and culture. In light of these point, the only thing in common seems to be a growing suspicion of rationalism.