They’re often used interchangeably to refer to ‘city law’ firms which tend to be the large firms in London. Corporate law, however (and confusingly), is also an area of law relating to the sale/merge/purchase of companies. The jargon is all pretty muddled but I would just take city/corporate/commercial law in your context to refer to large (often) London-based firms. By large I’m referring more to the sorts of deals they do - there’s plenty of smaller ‘boutique’ firms which would still be classified as practicing commercial/corp law.