Well, these words generally mean the same (and often translate the same, i.e. "expensive"), but they are used somewhat differently...
Aspect 1 - idiomaticness: If you go to the shop/supermarket/restaurant and want to complain about its products' prices you will use "drogi". The same with tickets.
Aspect 2 - emotional charge: Using the word somewhat implies that as for you the price of the product seems high. Maybe you believe the price of the product is higher than its real value. Maybe it's just you don't have much money and you can't afford to buy such a product. "W tym sklepie owoce są drogie."
Aspect 1): Used with more abstract/general notions, like ventures, projects, actions : "kosztowne przedsięwzięcie"
Aspect 2: Sounds more formal and emotionally neutral. With this word you state that the product has a considerable value, but you rather believe it's not overrated or too much...