Here is a list to start with. While the Austrians have always been an 'alternative' school, Chicago emerged from Keynesian shadows and has become mainstream today. Austrians have interesting perspectives on capital-based macroeconomics and boom-bust cycles while Chicago economists have contributed to a whole range of topics including macroeconomics, price theory, asset pricing, etc.

Austrian Chicago
Believes money should
be controlled privately
(issuance and circulation).
Likewise, banking should be private.
Believes in central planning
of money supply and a
somewhat ambivalent view
of the track record and
performance of the
Federal Reserve System.
Understands that human
behavior can sometimes
be irrational. Perfectly comfortable with
the boom-bust cycle
of the economy and asset prices.
Strongly believes in
the efficient market hypothesis.
In finance theory, there is no
concept of 'asset bubbles'.
Where prices veer away
from fair value, they correct instantly.
Favors a gold-backed
currency system. Even if the
government controls the money supply,
it wouldn't be able to fund wars
in the absence of
restricted supply of gold.
Precisely due to this handicap,
Chicago advocates no
such restriction on
the part of government
thereby encouraging
intervention of all sorts.
Austrian economics believes
in a set of logical principles that require
no empirical testing. That import tariffs are
bad need not be tested with
statistical data from several countries.
All that needs to be shown is that
consumers are worse-off and their
subjective value of needs
is not better satisfied.
Chicago economists believe
in rationality and optimal choices.
Theories are tested against
empirical data. New theories are
developed based on scientific data.
Unlike Austrians who view
rationality is bounded by
the current circumstances,
Chicago economists view
rationality as the
best/optimal decision.

Needless to say, both schools believe in free markets and consumer choice. Significant differences exist in the areas of monetary policy and the role of a central bank.