I'm no grammar expert, but the difference in your two sentences is simple. Won't implies that your computer is currently not starting and wouldn't implies that it did not start sometime in the past. Yesterday my car wouldn't start.
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Ivanhr1: My computer won't start. ( =every time when I try to switch it on)2: My computer wouldn't start (I tried to switch it on but it wouldn't start; said of a past event)The second is similar to "My computer didn't start" but gives the impression as if the computer refused to start.Thanks everybody.