Usually a bigger core means more headroom before the transformer saturates and distortion occurs.
But I think going way bigger can be detrimental to the high frequency response.
But I hope our transformer experts chime in an can explain that in detail.
Same number of turns on a bigger core means more wire which means more resistance which means more noise. Longer wire means more capacitance which means poorer HF response. Same turns on a larger core does mean the level required to saturate the core is higher but this is only really important at frequencies below 100Hz.