The difference is purely grammatical. 'Advice' is an uncountable noun. It's indivisible. That's why you can say 'I need advice' or 'I need some advice' ( but you can't say 'I need an advice') - just as the same as you'd say 'I need water' or 'I need some water' ( but not 'I need a water'). If you want to make 'advice' divisible, you have to say 'a piece of advice', just the same as you have to say a glass of water, a bottle of water or a litre of water. Glass, bottle and litre are the units we use to quantify amounts of water, and 'piece' is the unit we use to quantify advice.

Hi Dima, Good question. A 'piece of advice' is simply a single act of giving advice. "Let me give you a piece of advice about living in America." "Advice" refers to the noun meaning to offer direction or opinion. "I give advice to students quite often. To each of them generally just a piece of advice." I hope this helps, Stephen

It's like "water" and "glass of water".