East Coast or West Coast Swing?

That depends on your preferences....East coast is a circular type dance, you tend to move in a circular pattern. West Coast is a 'slotted' dance, designed to use a smaller floor space when big ballrooms disappeared after WWII.

In WCS, the man moves basically side to side as he conducts the woman 'down the slot' or along a track. There is a walk walk step instead of a rock, step, as in East Coast. You can dance thousands of patterns of WCS on a section of floor a little bigger than a door, but you need more room for ECS.

ECS has 'lilt' or a little rise on the 'and' beat steps. WCS is smooth, and should look effortless. ECS is usually a little faster.

Different music styles as well.

I'll also add that there are different musical styles underlying the dances. East Coast Swing works well with faster 50's rockabilly (early rock and roll) or neo-swing. West Coast swing adapts much better to contemporary, slower tempo dance music you might encounter on a bar juke box (if you can find one). West coast swing also has a more rounded style of movement as compared to something like Hollywood style Lindy hop, which stays more slotted.

Glenn's answer is a pretty good one. The reason for the development of Western swing is in some dispute, and it's likely the slot was originated for Hollywood's camera angles in a day when movie cameras weren't as capable of moving to follow the action as they can be now. Most people learn the older version first, finding it gives a good foundation for learning the western later.