Communism, as practiced in the USSR and China, means the State is in total control of all the production and services, and there is no private business at all.
Socialism expects a mixed economy. The state looks after essential industries like water, fuel, education & transport and takes care of the welfare of those in need. Otherwise people can run their own businesses, employ other people, and make a fortune, as long as they pay their fair share of taxes.
We have never had communism here, but for quite a while we had a successful socialist Keynsian economy, until Thatcher changed the rules.
Corbyn is not and never has been a communist of any sort. But Blair and New Labour were only slightly socialist, since he carried on with Thatcher's economics mainly. We are just getting back to the mainstream left now .