能 can indicate something that you are able to do naturally, for example:

I can eat 3 bowls of cooked rice.
我能吃三碗米饭。
Wǒ néng chī sān wǎn mǐfàn.

能 can also be used to ask for or give permission, for example:

Can I smoke here?
我能在这里抽烟吗？
Wǒ néng zài zhèlǐ chōuyān ma?

No, you can’t.
不能。
Bùnéng

会 means that you have acquired certain abilities or skills, and that you have acquired them through learning, for example:

I can speak Mandarin.
我会说普通话。
Wǒ huì shuō pǔtōnghuà.

I can cook.
我会做饭。
Wǒ huì zuò fàn.

Can you drive?
你会开车吗？
Nǐ huì kāichē ma?

“会” also indicates what might happen in the future. For example:

It won’t rain tomorrow.
明天不会下雨。
Míngtiān bú huì xià yǔ.

Welcome to have my 1-ON-1 lessons on http://www.verbling.com/teachers/dawei 🙂