担責任 is a verb. 負責任 is both a verb and an adjective. For example, 他可以担責任/ 他是個可以担責任的人/ 他可以負責任/ 他是個負責任的人

没错是一样的！

被动与主动

On usage, 担责任 is "more commonly" to describe "take the responsibility for other people", on the other hand负责任 "take the responsibility for yourself [ as the problem is created by you]....But there is no fixed rules...many people use it freely.......

OK.you can understand it in this way.The first one you can understand in like,“alright,the trouble has been made,and its all my bad,dont worry i got this,i will 担责任,and the later is more like"i am so sorry about this shit happning on you,i will totally be responsible to you, i will 对你 负责任" hope it can help you