How might a socialist and a capitalist government differ in treatment of the problem of unemployment?

There are no purely socialist or capitalist countries.

In a purely socialist one, there is no unemployment, the state will find something for you to do.

In a purely capitalist one, there is always some unemployment, but small and transitory. Private charities would make up for what shortfalls took place.

In mixed economies, there are a mixed bag of responses. Unemployment benefits help. As do various aid programs, such as food stamps and utility assistance plans. The more "socialist", the more such programs are likely to exist.

The more capitalist, the more such problems as unemployment are unlikely to exist, with charities taking up the slack.