際 vs. 時: What's the Difference?

The meaning is almost the same but slightly different.

際（さい）

It is honorific.

Strong purpose. (It usually means having a great opportunity.)

Always used as Written language. (But it can be used in spoken Japanese.)

It is a standard term usually used on business occasions and official documents.

この際思い切って家族みんなでアメリカに引っ越しませんか。

In this case, "際" means opportunity.

時（とき）

Always used as an Oral language.

It is often used for things that happen by accident.

寝ているときには地震があった。

You can't predict that an earthquake will happen. So, in this case, we use "とき";

Summary

"...時" usually means that other things happen in parallel or a specific state occurs at the same time;

In addition to the usage of "...時", the "...際" is more solemn in tone than "...時" and has the meaning of "opportunity."