QUESTION: 
Is there a connection between having higher mountains on Mars and the fact
that atmosphere is thinner? Or is the thin atmosphere due to the smaller
diameter and gravity of the planet?

Would this mean that a volcanic eruption would "shoot" higher due to the
smaller gravity?

ANSWER from Bruce Jakosky on September 15, 1997:
The fact that Mars has taller mountains than the Earth and a thinner
atmosphere probably are unconnected.  The height of the mountains depends on
things going on in the interior--the density contrast between the mantle and
the crust, the lack of plate tectonics, and so on.  The thickness of the
atmosphere depends on the amount of gases incorporated into the planet
during its formation, the history of outgassing (although this connects also
to the history of the interior), and the subsequent processes by which
atmosphere is lost to space.

The height of eruption of a volcano has two components--how fast does the
material shoot out of the vent and how buoyant is the hot gas in the
atmosphere.  Even on Earth, the first can take material to only a kilometer
altitude at most.  The high volcanic plumes that reach into the stratosphere
rise that high because the hot gas and debris together are less dense than
the surrounding gas; as a result, the plume rises like a hot-air balloon. 
On Earth, plumes can rise to 45 km altitude.  On Mars, they might rise to
similar altitudes or to just a little less (because of the less-dense
atmosphere).

Are there active volcanoes on Mars?  We haven't seen one, but there still
may be active volcanism.  Volcanism has occurred recently (within the last
few hundred million years), based both on photogeological evidence and on
measurements within the martian meteorites.  If it has ceased completely,
then it has just done so very recently.