With an S-Corp, you pay your owners a reasonable salary. $50-70k is reasonable for our field according to our CPA. Any profits or losses from the business pass through to your personal income tax via a Schedule K1. The profits are taxed like normal income, they are not taxed as capital gains.
The tax savings come from the fact that the distributions you draw from the profits of the company are not subject to FICA (medicare, social security). This saves you upwards of 13%.
If you draw more than your profits for the year (say you run an accrual method of accounting, not cash, so you recognize revenue after services have been fully rendered, rather than when the cash comes in and you can have a large bank balance but no profit for that year), the excess is essentially double taxed with capital gains. A painful mistake, so track your AAA appropriately :)
Taxes paid via withholding are treated as if they are paid in through the course of the year which avoids estimated tax payment penalties.
We have a pretty sophisticated business intelligence platform written in R that gets us estimated revenues within 10%, so I know how to budget through the year for taxes owed. I just pay my known quarterlies and keep enough in the bank to handle the remainder at year end.
In regard to your question about why your misconception is common, no idea. Just a quick google shows lots of accurate descriptions of the benefits of an S-Corp. Maybe a misunderstanding with your accountants?
After moving to a larger (and much more expensive) accounting firm, and learning from our previous mistakes, I've had a much more pleasant time dealing with this stuff. It's all arcane and stupid when it comes down to it, but you have to learn it to effectively manage your tax burden. If you don't actively think about it you get absolutely screwed on occasion, as it affects everything about your business. And the mistakes get expensive as your revenue scales up.
This is getting totally tangential, but you can probably hardly imagine all the crazy rules regarding taxation in the case of a sales event. Operating as an S-Corp is fine and dandy, but creates an additional hurdle for getting investment and for sales. S-Corps can only be owned by individuals, so an acquisition or an investment means restructuring, but you are restructuring after a planned purchase, which says certain things about intent which makes the restructuring likely a taxable event in and of itself.
I strongly strongly strongly recommend talking to an experienced business lawyer, and having them recommend an experienced business tax planner. This isn't schemey tax avoidance stuff, this is fundamental planning so you don't get a surprise $200k bill at some point!
"Then, any remaining profits from the company can be distributed to the owners as dividends, which are taxed at a lower rate than income."
Though it doesn't specify what that rate is. Most of the other results indicate that the additional distributions beyond salary simply avoid self-employment tax. So it would appear that distributions beyond salary incur less tax, but are taxed higher than the current capital gains tax rate. It's also important to note that distributions are taxed - if you leave the money in the company, it's not taxed until it's actually distributed to the owners.
Shareholder distributions then add (or subtract if you lost money) to/from shareholder individual incomes. It is taxed at the same level as other income.
Also others are correct in saying that an LLC can elect to be taxed as an S-Corp. S-Corp is a concept in the tax code, not a form of ownership itself. That is, if you create a corporation with the state, that corporation is just a "corporation", not an S-Corp or a C-Corp. Your corporation will be classified as an S-Corp or C-Corp depending on how you elect to file taxes with the IRS. It's similar with an LLC, which you create with the state, then can elect what tax treatment you will use with IRS: typically as sole proprietorship, partnership, or S-Corp.
With an LLC, all profits fall through to your income and is taxed as such.
Edit: revenue -> profit