I've recently started to learn about Chern and Segre classes, and it seems to me that they are  very similar, sharing the same important properties and having closely related definitions.

Fulton's book starts by defining  Segre classes, and then defines Chern classes. 
My question is: what are the advantages of Chern classes over Segre classes (or vice-versa)?
Are they really very similar, or is there some conceptual difference?