It's natural. The raster images in the browsers are shown pixel-to-pixel unless the soft use any scaling algorithms. Mailing services do this. So, if you export your logo with 300PPI, it will be scaled to a small size as needed in your signature, and look bad due to the sensitivity of raster images to scaling. Export it with 72PPi. It will look bad in big size but very well in the small size in the signature.
As far as I know, most of the mailing services don't allow to use of SVG (vector image) format, but if yours does it, you can try to cave your logo in SVG.
But, as Monika said, these small images generally just clog up the letter attachment set and annoy the receiver.