Question: I keep reading about joining a small start-up, keeping it up until it grows and then reaping the rewards.
However, the way I see it is that investors and shareholders will be reaping the rewards, while overworked employees will get a promotion at best, and they could have gotten that promotion in many other companies. Some make millions, some make 10K more per year.

Shouldn't "reaping the rewards" include seeing your equity/shares become much more valuable? If so, should one join a start-up only when equity is written in the contract?

Answer: 
There are plenty of benefits to joining a startup without equity.

1. Rapid promotion
2. Experience doing more things and having more responsibility and ability to set direction than you would at a larger shop
3. Exposure to startup people, both in your company and funders, for your next startup

None of these will make you a millionaire, however. If you want to "cash in," then yes, you want equity and you want it in writing. 

I have plenty of friends that worked for one startup or another, and even if they were "there early," when the founders sell and cash out, you don't get a handout out of the goodness of their heart - you get nothing, if you have no clear equity agreement.

So while there are reasons to work for a startup even if you don't get equity, if you are specifically looking for the cash-out part of that life, then you absolutely need a formal equity agreement - though even then, I've seen buyouts that manage to somehow screw minority equity holders, so it's no ironclad guarantee. As a deleted user mentions in his answer here, making sure you have a percentage or otherwise legally-enforceable non-dilutable claim is the only way to ensure you win.
If it is a competitve salary doing work you enjoy then why not. 

How can you assume will be overworked? 

What is wrong with promotions? If a small company grows there are more opportunities for promotions.

And some investors make nothing. If the start up fails you get to keep your salary. 

By that logic should you take corporate job without shares?

In another question you complain about lots of politics and manager taking credit for your ideas. A small start up is not likely to have those problems.

The stated question is take a job without equity. And the answer is simple. If the job, salary, and benefits is competitive then why not? You are not getting equity at a large company. Evaluate the offer for what it is. 

A start up is not necessarily a sweat shop. It may be well funded and they already have the core competency and you are just a valued well paid employee. 

If they need your skills to develop the core competency then yes they are going to offer shares. 

If they offer a lower salary and shares then you have to compare that to another a higher offer with no shares.