It makes a big difference. You can still get some nice sound, but it's not like you've got a powerful amp, so obviously you choose your headphones accordingly. Will you be listening to Spotify, or what? And is it for on a train, walking, at home, or what?
I see you budget is £300. What is your source?
Noise cancelling works when there is a constant drone you are trying to cancel out. Perfect for regular flights. Decent for train journeys. Rubbish for most environments.
Over ear, open, are generally better (all else equal) at giving a good sound...