厚生, 福祉 and 福利
First of all, 福利 and 厚生 are almost outdated words except in legal terms or a combination of 福利厚生 ("fringe benefits" according to WP). 福祉 is the most common word covers "welfare" in every situation today.
福利 particularly refers to benefits or services one can gain from welfare system. And 厚生 literally means "fulfilled life" or "life enrichment" (or "high QoL" if you prefer), which is used as a fancy word in place of "welfare", much as like we call the judicial administration "Department of Justice".
It's a proper name given to Japanese social welfare program officially translated as "Public Assistance". You cannot consider this as a term indicates any kind of generic welfare systems.
A general term translates "public assistance" or "social assistance" seems to be 公的扶助.