I like the Woodsman III medic because he is a better medic. The healing rate is higher with Woody III than with Combat I and Medic II; though, he doesn't heal in adjacent tiles. 27(?) standard XP gets you Woody III, Combat I, and Medic I, which is plenty good enough for me and is superior to the healing that any Chariot can ever achieve.
The trade off is that your Woody healer is slow and winds up staying behind with the beaten up Cats and misses out on the chance to grab an XP or two at the next city. If you're a Spies plus Horseman-cum-Cavalry style attacker, then the Chariot is certainly the way to go.
I do admit that if you don't have Agg, Cha, or Jaguars, it is difficult to create a truly great healer unit, and in that situation I often do wind up using a Chariot with my GG and just trying (and often failing) to get a standard unit to 10 XP and Woody III on its own to hang back with the wounded siege units. With any of those three Characteristics however, I think that the Woodsman line gives you a better healer sooner.