In addition to what k said without actually mentioning the Pen Tool, I would also suggest your using it.
In this case, you can choose a full recreation of the asymmetry and (Pen) trace each, or you can choose either or combine parts from both and reflect for symmetry, or anything in between.
And whichever of those ways you can choose to follow the actual shape(s) more or less strictly/loosely, including the (variations in the) thickness of parts.
Once you scan it in and bring it into Illustrator, go to Object>Image Trace. Or it might be active when you select your image on the options bar. Since this is a pretty clean image, it might trace well immediately. Or on the option bar, there is a little place next to default that you can click to bring up options. There you can pick from various modes that may work better rather than the defaults, like Black and white logo. You may also want to expand Advanced and under the options there ignore the white.
Once you have vectorized the file, you can do a file save or you have various options depending on what you want to do with it. Save as a PDF for example, export as a png. If you tell us what you want to do, we can give you more guidance.