To distinguish means to tell apart. If you win the lottery, you’re going to need to learn to distinguish between people who really like you and people who like your money.
To distinguish is to discern, which means to perceive or recognize the way something differs from what’s around it. It takes discerning taste buds to distinguish between Coke and Pepsi. Distinguish also means to separate yourself out from the pack, usually in a good way. You might distinguish yourself at a Comics Convention with your encyclopedic knowledge of Krazy Kat.