The simple solution to a problem can very often be the right one, but a simplistic one can never be anything but wrong. If you say it's simplistic you mean, or ought to mean, that the difficulties involved have been underestimated - oversimplified, in fact. Cross out and do again.
Ms Brown is by no means the only person to have fallen into this trap, which seems to be claiming more and more victims these days; and it can happen to the best of them. Even the Guardian's Ian Aitken fell into it a few years back (I remember this because I happened to be collecting malapropisms for a book at the time), when, in an article about proportional representation, he wrote of "the real crunch in this otherwise simplistic matter of fairness versus unfairness". A subject can't be simplistic in itself, only people's approach to it. He must have meant "simple".
It's usually a mistaken desire for a bit of extra gravitas, a bit of weight, that makes people prefer the longer word to the shorter. Etymologically there's not much excuse for it this time.
Leaving aside the professionals like scientists and linguists, anyone who can be described with a word that has an -ist or an -istic on the end of it is likely to be suffering from an ism. Thus an atheist is a person who keeps on insisting that there's no god; an evangelist is bursting with a message; a materialist is obsessed by worldly goods. And a simplist (the word exists) sees everything in black and white. Realists, admittedly, are an exception: they see things as they really are and should presumably be given the credit for it. So perhaps a simplistic person might be given the credit for seeing how simple things are? If only they were. (Even realists, come to think of it, can be rather tiresome at times.)
At least one paper described the killers as having run amok, which reminded me of the old Daily Telegraph style-book - long out of print now and much prized by the few who still have a copy - one of whose instructions read: "Only Malays run amok". This was indeed true for a long time, amoq being a Malay word for a frenzied warrior, but it was already being applied to others a century before that style-book was compiled.
If enough people go on using simplistic when they really mean "simple" then that is what it will mean, and all the style-books in the world will be powerless to stop it. But I rather hope this won't ever happen, because if it does we will have lost a useful word, and will have to find another. Simplificatory perhaps? That should be long enough for anyone.