As wineis.. was getting at, distinguishing the two can be difficult, if only because of the great range of styles in CCR. It is also possible that both wines can be made of exactly the same grape constituent percentages. That said, they do call their particular clone of sangiovese Prugnolo Gentile in Montepulciano, so there are probably some clonal variations to account for.
I would suggest, however, that the real distinguishing element is the taste as effected by terroir (i.e. soil and climate).
Montepulciano has lots of clay in its soil, whereas Chianti Classico is more stony. I'm uncertain of the exact mechanisms by which soil types affect taste so I won't get into that, but I will say that in general, I find the wines of Montepulciano to be softer, less heavily structured, and not as intense as their Chianti Classico counterparts. Apparently this is due to clay holding moisture and remaining cooler than the faster draining stony soils in Chianti, because again, grapes and winemaking are generally similar.
An excellent demonstration could be made of comparing wines like '02 Avignonesi Vino Nobile di Montepulciano with '02 Badia a Coltibuono Chianti Classico Riserva, or '01 Corte alla Flora Vino Nobile di Montepulciano Riserva with '01 Il Molino di Grace Chianti Classico Riserva.
Hope that helps!