AE does not have variable strokes liek AI. You need to come up with a different way of creating your strokes, possibly using trapcode 3D Stroke for instance. The rest we can't really know without seeing a screenshot/ reference frame of your logo.
You can use a track matte or Set Matte on a wider enough animated shape layer path to give you the various widths. You could reverse the process by drawing a path over the filled shape in an Illustrator file and use the Animated Path as a track matte for the layer. Those workflows are easiest if you set up a layered Illustrator file with the shape you want to reveal on one layer and a simple single-path on another layer that traces over the center of the shape you want to reveal. Do not add a brush, just a simple path. Import as a composition, convert only the Path layer to a Shape layer, add Trim Paths, and set up the matte. I use that workflow all the time. Laying it all out properly in Illustrator is almost always faster than trying to create the path in AE. It is one of the few times I convert vector layers to shape layers.