Slice suggests something which has been cut (with a knife) whereas "piece" suggests something pulled or torn or left over such as: I cut the loaf with a bread knife had a slice with peanut butter. All that was left of the loaf was a small piece which I broke in two and devoured with peanut butter.

In my opinion, slicing the bread means cutting the bread by knife regularly, but piece of bread means tear a whole bread into pieces by hand haphazardly.

The word "piece" is more useful as a general purpose counter for non countable nouns, for example: a piece of gum, a piece of advice, a piece of the action, a piece of chocolate, etc.