In British English, welfare and benefits can be the same. Both mean "help from the government." In American English, welfare means "practical or financial help that is provide by the government for people or animals that need it." Benefits are things, like medical insurance and paid vacations/holidays, which come only from the company you work for.

In the context of providing assistance to the underprivileged, I see 'welfare' as being more of an American term. Americans talk about 'welfare checks' and being 'on welfare' and 'welfare recipients'. In Britain, these terms aren't used. In Britain we talk about being 'on benefits' or 'receiving benefits'. The only time that 'welfare' is used in this context in British English is to refer to the establishment of the 'Welfare State' in the mid-20th century. We don't use it to refer to 'help from the government'. This help is called benefits, not welfare.