They couldn't be much more different, really.
Ares was part of a dichotomy of war gods, comprised of himself and Athena. While Athena represented wisdom, strategy and generalship in war, Ares represented base aggression and the brutal side of war.
As such, he wasn't particularly well regarded by the greek populace, and instances of his worship are relatively sparse. Usually, when he appeared in myths, he served as a foil to be conquered by a smarter (and better respected) opponent, such as his defeat by Athena in the Trojan War or his humiliation at the hands of Hephaestus.
His name is believed to be from the word "are", which means: "bane, ruin", so from his start there is a clear indication of people's intended relationship with him.
Mars, on the other hand, is among the most important gods of the Roman pantheon, and second (to Jupiter, of course) member of the Archaic Triad, and the most significant of the gods of the Roman military. He is a protector (Mars Quirinus) and father (Mars Pater) of the Roman empire. Soldiers swore oaths to him (Mars Gradivus) to be valorous in battle.
Edith Hamilton's "Mythology" has this to say:
The Romans liked Mars better than the Greeks liked Ares. He never was to them the mean whining deity of the Iliad, but magnificent in shining armor, redoubtable, invincible. The warriors of the great Latin heroic poem, the Aeneid, far from rejoicing to escape from him, rejoice when they see that they are to fall "on Mars' field of renown."
An interesting take on the matter, is that perhaps, in terms of function, Mars and Ares aren't really the most directly comparable, but rather that Mars is more directly related to Athena, and that the inheritor of Ares's function is the goddess Bellona, as posited in this thesis:
Accordingly, Roman religion recognized a male warrior god with both agricultural/fertility and martial aspects. Roman religion attributed a much more defensive mentality to Mars than the Greeks attributed to Ares. With no push for imperialism in early Rome, the female goddess associated with rage (Bellona) did not emerge into prominence until Sulla.
In the observations made up to this point, a chiastic inversion occurs between Greek and Roman religions. The inversion can take two different forms. First, the Roman gods have switched genders from their earlier Greek counterparts...
War Gods in Archaic Greece and Rome, Tyler Krentz