  Tuffley and Steel (1997) proved that Maximum Likelihood and Maximum Parsimony
methods in phylogenetics are equivalent for sequences of characters under a
simple symmetric model of substitution with no common mechanism. This result
has been widely cited ever since. We show that small changes to the model
assumptions suffice to make the two methods inequivalent. In particular, we
analyze the case of bounded substitution probabilities as well as the molecular
clock assumption. We show that in these cases, even under no common mechanism,
Maximum Parsimony and Maximum Likelihood might make conflicting choices. We
also show that if there is an upper bound on the substitution probabilities
which is `sufficiently small', every Maximum Likelihood tree is also a Maximum
Parsimony tree (but not vice versa).
