The weather is windy and the wind is blowing outside. I am sitting in front of the TV, but it …………… turn on and I don’t know what I should do to fix it.
closed as not a real question by jwpat7, Lynn, coleopterist, Mahnax, Andrew Leach♦ Sep 9 '12 at 10:02
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They mean different things. They may or may not "work". Depending on what the speaker wants.
The will of It won't turn on is the Deontic Modal will, meaning (be) willing (to). Since television sets are not capable of willing, this must mean failure to turn on has been diagnosed as a persistent "habit" (i.e, tendency) of the TV. Hence there is an invoked history. And a personification of the TV set.
The do of It doesn't turn on, on the other hand, is simply ordinary Do-Support, with no special meaning at all, and therefore -- by contrast with will -- it carries the sense of a first encounter, an alls-I-know-is-it-didn't-turn-on event.
Aside from that, no difference in this case. Other cases, of course, will vary.