负责usually means take the consequences for sth or behavior,usually as a verb
For example
1、这个项目由你来负责(it means you lead and complete it)
2、你要为这个项目的失败负责(it means you must take the consequences of failure)
有责任 usually means a kind of obligation,you should not evade,you should do it in the range of your ability
for example
1、做为父亲,你有责任让自己的孩子接受教育(you should let your kid get an education because you are father)
2、我们有责任帮助那些有困难的人(we should help other people ,because we have this ability)