An ice axe is primarily for self arrest if you slip on a snow slope, although can be used to steady yourself on a slope, and for cutting toe holds in hard snow.
I went up Fairfield last February: snow conditions were excellent but the snow was hard - so hard that it was difficult to kick a step, despite wearing heavy winter boots, and I did use the axe to cut a few steps in a steep gully. Whilst on the walk, I was surprised and a little dismayed to see so many people using poles instead of an ice axe - they would have been in trouble had they taken a fall.
Practising self-arrests in safe locations is also fun and confidence-building
I remember being on Broad Crag in the ice and chatting to a fell runner from Keswick about (cough) 25 years ago. He ran down to Esk Hause, and fell and instantly self-arrested several times, and continued quite happily - presumably all the way back to Keswick.