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That's happening because the logo is filled with a gradient. And Trace Bitmap can only trace individual, or flat colors.
By any chance do you have a version of the logo without any color, or with a solid color? If so, you could trace that, and then apply a gradient to it, in a 2nd step.
If not, you could use the Pen tool, and trace it "manually". And then apply the gradient. I'd be happy to help you learn how to use the Pen tool, and Gradient tool. You might even have the gradient saved separately, somewhere, if you have had any plans to trademark the logo?? Or actually, I think there are 2 gradients (one for the logo and one for the drop shadow).