Visa Sponsorship is essentially a company/work place sponsoring your visa (haha sounds obvious I guess), but it's basically the employer taking responsibility of hiring you.

Regarding the Certificate of Elgibility, it's essentially a process where it speeds up your visa application process (AFAIK). For you to get one you actually need sponsorship from a company... EVERYONE does. Even if you are in Japan for study longer than your intended tourist visa you need visa sponsorship from your school/language institute for your student visa application. COE's are essentially documents from the school/employer etc certifying that you are indeed in Japan for the intended purpose of working/studying, which helps immigration then process your applicaton quicker than normal.

As far as I know you need both (Sponsorship for your Visa and a COE) for you to get your Visa Application approved if you plan on working.