I'll add to Wayfaring Stranger's comments. In fact most of the time you would be able to see fewer stars in the night sky of Mars, than in a good dark night sky on Earth, because of dust obscuration.
Even in favourable conditions, the optical depth of the Martian atmosphere is usally somewhere between 0.5 and 1 per airmass. (Petrova et al. 2012; Lemmon et al. 2014) and is nearly wavelength independent. This corresponds to a reduction in flux to between 37% and 60% of its value above the atmosphere. This compares to a typical optical (V-band) extinction of around 0.1 magnitude at a good site on Earth, which allows 90% of the flux through. There are times when it can be much worse than this on Mars.
This means that the limiting magnitude on Mars would be somewhere between 0.44 and 0.96 magnitudes brighter than it is on a dark site on Earth and that all the stars would be fainter by these values.
As a rough guide to the effect this has I took the Hipparcos catalogue and made a frequency histogram of V-magnitude of the stars. If we define a Hipparcos magnitude of 6 as the limit you can see from a good dark site on Earth, then you would see 4559 stars (that's over the whole of the sky in both hemispheres). If that limit was reduced to 5.56 or 5.04, then this number decreases to 2745 or 1560 respectively. Thus there would be a reduction in the number of stars you could see by typically between a factor of 1.66 and 2.92.
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