Cut the price in half and by half mean the same thing. You take whatever the price is now, and you reduce it by 50%.
However, cutting the price to half, may have a different meaning. That would imply that the price is reduced to 50% of whatever the original price was. So if the item was already discounted by 20%, it would now be discounted by 50%.
To clarify this, lets assume we have two items, one with no discount (A) costing £1, and one with a pre-existing 20% discount (B) with an original price of £1.
If the price was now cut by or in half:
A now costs £0.50.
B now costs £0.40 (0.8x0.5)
If the price was cut to half:
A now costs £0.50
B now costs £0.50