I dance and teach both Lindy and what I call vernacular East Coast Swing.
Lindy Hop is taught with an eight count basic called a "swing out". The lindy circle oft referred to in this post is a method for getting into closed position in order to facilitate tuck turns, side by side Charleston moves, etc. Depending on the beats per minute, lindy stays at eight counts or drops to six simply because there isn't time to do anything in eight. It has evolved quite a bit from its origins, which was more of a circular Charlestonesque rock-step kick turn rock step kick turn. That old fashioned Lindy is fun, but frenetic.
East Coast swing breaks into two categories of its own, triple step East Coast and the vernacular East Coast. You might call it "street style" as I read earlier. Personally, triple step East Coast is for Ballroom Dancers. Any song slow enough to make triple step East Coast comfortable for me I'll be dancing Lindy Hop. As the BPM rise and catch up with my injuries, general out-of-shapeness and lack of experience/practice with faster Lindy Hop, I'll dance single-step East Coast.
Both East Coast Swings are taught in a broken rhythm, (6 counts when the music has 8). In East Coast the dancers face each other, usually with a two hand hold. The basic travels side to side with a rock-step backward.
A Lindy basic, the swing out, begins with the two dancers facing, lead's left hand held to follow's right. The lead rock-steps, they come together and turn into closed position, continue the turn and the lead sends the follow back out to her original position as they complete the "circle".
They aren't dissimilar, particularly since one did indeed develop from the other. I believe the order of progression goes Charleston -> Lindy -> West Coast | East Coast.