Here's my understanding:
Shared send is like a traditional mixing service. You send bitcoins, and your destination address receives bitcoins that came from someone else. No link in the blockchain between your input and output if all is done correctly.
Shared coin uses coinjoin. You send bitcoins in a transaction that includes a bunch of different people sending bitcoins as well. The result is one big jumble where it's difficult to tell which inputs and outputs are related. The link between your input and output in this case still exists but is obfuscated significantly. This option also has no fees besides the network fees.
Thanks! After some searching that is also my understanding. The SharedCoin website is a bit confusing and makes SharedCoin appear to just be a clone of Send Shared. From the SharedCoin website:
"Coins send with shared send will be matched up with another user. When a match is found your coins will be swapped breaking the transaction chain from your own wallet. Coins will be swapped with multiple users making the chain even harder to follow."
The website for SharedCoin is using the description from Shared Send, and a coinjoin mixer is never mentioned.