Something like this maybe? The Bevel deformer is just there to give the object some depth. Thismaybe.zip

Like explained, a sweep is to give a spline (you can not see/render a spline, there are no polygons to render) thickness by using another spline. Forget the sweep (but if not): Take an Arc, sweep it with a Circle, in the sweep, open open the little triangle for details, and you can play with the curve for scale. Other option:  Use a spline Mask and Extrude that. A Spline Mask is like a Boolean for splines. So you need your logo spline (Path.1 in your file), and something else, like a Rectangle. Make a Spline Mask and make both Splines a Child of the spline mask. Depending on the order, set the spline mask to A subtract B  or B Subtract A. You can now animate the Rectangle to change the shape of the spline. (be careful to not move in the wrong axis) And now you can drop this Spline Mask in an Extrude Object, and you have your logo animated! Option 3:  Use a real Boolean! Put your Extruded Logo under a boolean, and put a Basic Cylinder under that as well. No animate the Slice of that cylinder. Booleans are , but your case it should work pretty sweet.

You haven't done anything wrong. Where things don't match is a seam. With UV mapping you're trying to hide seams as much as possible.