Both mean "nothing" as far as I understand, so what is the difference?
Russian declension is a very complex system. Its grammatical cases (падежи) sometimes can be very tricky and confusing, however there are some rules that can help you to understand them.
In your case, you actually have the same word in two different cases. The first one, "ничем", is an instrumental grammatic case for the main - so called, nominative - case "ничего".
All six grammatic cases in order:
Именительный падеж/nominative case:
Родительный падеж/genitive case:
Дательный падеж/dative case:
Винительный падеж/accusative case:
Творительный падеж/instrumental case:
Предложный падеж/prepositional case:
(Exception) Ни (о/в) чем
It's totally OK to say: "Он абсолютно ничего не сделал"/"He has done absolutely nothing", while "Он абсолютно ничем не сделал" is incorrect, even though the majority of Russians would understand you.
Number of cases in Russian language can differ in different textbooks, but I cannot mention any noun that would not have any of these grammatic cases.
ничем is declension of ничто in instrumental form (see Russian_declension). Example: negotiations ended with nothing - переговоры закончились ничем.