From a UK perspective, I would say that you can use either when referring to somewhere you play team sports. 'Football pitch' and 'football field' are both correct. However, 'pitch' implies the area is more professionally set up, with lines marking out the pitch - you would be unlikely to refer to the 'football field' at Wembley; you would use 'football pitch'.
Some sports (generally individual, or small team sports) are played on a 'court', not a pitch. E.g. tennis court, squash court, volleyball court. But you would not say 'football court'.
'Pitch' is also a camping term: at a campsite you would pay for somewhere to 'pitch' your tent (a marked-out area of land), and you would then 'pitch your tent' (erect it).