The two mean similar things alone themselves, but they do not share collocations. 承担, meaning to undertake, to shoulder. It often goes together with responsibility. 担任, is to assume, to hold. It is usually used with a job title. The confusing thing is in English we can take both "responsibility" and "a post". The two collocations are acceptable in English. But as you can see, that's not the case in Chinese even though 承担and 担当 seem to mean the same thing (to take) literally by themselves.

承担责任，担任后面加某个职务