Verified answer

lit. please do. onegai, shimasu.

Onegaishimasu is only used after requests, in the case where you would ask someone to "please do" something for you. i.e. after ordering, if you are asking a favor.

Kudasai is less formal, and you can use it in the same instances, but there are also some cases where you have to use kudasai, such as after -te verbs.

I disagree with belie. Kudasai is not just restricted to after te form; it can be used like in the request:

kore o kudasai.

Onegai shimasu is a very versatile word and can mean many things depending on context. Here it is more like please.

Onegai has a lot more usage than kudasai, as kudasai can only be used with verbs conjugated in the ~te form and cannot be used by itself.

Onegai shimasu can range from "please" to "help me" to "go ahead" "nice to meet you." You can use it when asking for something, when ordering a meal, when introducing yourself (typically with yoroshiku before it), or when you're letting someone enter a doorway before you. It's a very useful thing to do.