Verified answer

Roughly,

ように > non-volitional verb, potential verb, intransitive verb > status

ために > volitional verb, transitive verb > action

I study to speak Japanese.

日本語を話すために勉強する．> 話す(volitional verb, transitive verb)

I study to be able to speak Japanese.

日本語が話せるように勉強する > 話せる(potential verb, transitive verb)

世界を平和にするために戦う > する(volitional verb)

世界が平和になるように祈る > なる(non-volitional verb, intransitive verb)

And "授業に間に合うために急いで家を出た。" is not correct. It should be "授業に間に合わすために急いで家を出た。" Though many people made this kinds of mistakes.

See this site for more information.

http://study.u-biq.org/66.html

ように means, like something.

ためにmeans, for something.

1) 彼のように歩ってください

1)Please walk like him

2)彼のために水を持って来てくれますか？

2)would you get the water for him

Sorry about the poor examples.

ように　＝　so as to

ために　＝ in order to

I think both are interchangeable, but "ために　in order to" emphasizes the purpose of the action than "ように so as to".

I hurried out so as to be in time for class.

授業に間に合うように急いで家を出た。

I hurried out in order to be in time for class.

授業に間に合うために急いで家を出た。

ように=like

ために=for