I always thought, from my Italian travels, that 'centro' pretty much meant 'downtown'. But perhaps it is more appropriately the 'town center', that is, the walking area devoid of cars. There may be a subtle distinction I need to remember.
I put 'are we going to the town centre today' and it was marked wrong. If you just say 'are we going to the centre' it sounds wrong - what centre is it referring to? We don't use the term 'downtown' in the UK.