With all due respect, this article really isn't that great.
Here's the real difference, stated succinctly:
LLCs and S-corps are pass-through entities that aren't generally subject to regular corporate income tax like C-corps are. (But there are exceptions, like NYC, which taxes S-corps as if they were C-corps.) Additionally, C- and S-corps can issue stock to owners and investors, while LLCs cannot, but S-corps are restricted in various ways that C-corps are not, like not being able to issue stock to foreign investors, having only one class of stock and no more than 100 stock owners.
VCs and Angels will not invest in an LLC, and the process to convert an LLC to C-corp is (or was, last time I checked) difficult, typically involving the formation of a brand new C-corp that buys the LLC and then dissolves the assets of the LLC into itself. I believe Joel once mentioned that FogCreek went through this process years ago, and it was not very pleasant. By contrast, it is trivial to convert an S-corp to a C-corp with one form (IRS Form 1120).
If you ever plan on issuing stock or taking outside investment, start out as a C-corp or S-corp. If you plan on running a business that won't (or can't) issue stock or accept outside investment (like a law firm or medical practice), then form an LLC.