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.