What is the difference between "kudasai" and "onegaishimasu"?

Both "kudasai （ください）" and "onegaishimasu （お願いします）" are used when making a request for items. "Kudasai （ください）" follows the object and the particle "o".

Kitte o kudasai. 切手をください。
Please give me stamps.
Mizu o kudasai. 水をください。
Water, please.

"Onegaishimasu" can be replaced with "kudasai" in the examples above. It sounds a little more polite. The particle "o" can be omitted in this case.

Kitte (o) onegaishimasu. 切手 (を) お願いします。
Please give me stamps.
Mizu (o) onegaishimasu. 水 (を) お願いします。
Water, please.

However, there are some situations when only "onegaishimasu" is used.

(1) When making a request for service.

Tokyo eki made onegaishimasu. 東京駅までお願いします。
Tokyo Station, please. (to a taxi driver)
Kokusai denwa onegaishimasu. 国際電話お願いします。
Overseas telephone call, please. (on the phone)

(2) When asking for someone on the phone.

Kazuko-san onegaishimasu. 和子さんお願いします。
May I speak to Kazuko?

For making requests that involve action (please listen etc.), the verb "te form" is added to "kudasai". "Onegaishimasu" can not be used in this case.

Chotto matte kudasai. ちょっと待ってください。
Wait a moment, please.
Ashita kite kudasai. 明日来てください。
Please come tomorrow.