They are almost the same. Κρύος means cold and Ψυχρός means cold but it has a sense of coolness or slightly chilly. Common expressions Έχει κρύο = It is cold, Έχει ψύχρα = It is cold or a bit chilly. You can also say κρύο αστείο meaning a joke with a bad sense of humour and you can say Είσαι ψυχρός meaning that someone is unemotional or unfriendly (the opposite of "warm people or person")

I'd like to add to the answer below that, as you said, you'll hear mostly "κρύος" in different contexts (either κρύο ανέκδοτο, or κρύος καιρός, or κρύος άνθρωπος). "Ψυχρός" is mainly used in its figurative meaning for people, rather than weather, now.