'Slice' emphasizes the whole from which a part is taken and that the process is done by cutting (with a knife or some other tool), so you often have a slice of pizza or pork. 'Part' emphasizes division so if you want to split the cake which you and your friend have made, you take your 'part' of it (perhaps with the help of a compass and your skills in geometry - a joke!). A piece of bread is most commonly used, but you could always make a heavenly loaf of bread with your friend and take your share/part of it (division emphasized). Bottom line, this is often a matter of meaning rather than grammar.