There is not any real difference in meaning most of the time, but the two phrases aren't necessarily interchangeable, either. In my mind, go without is a bit more of an action verb, while do without is more stative in nature. To go without (something) means to lack having it, that is, to actively NOT possess it. To do without something usually means to not have something, but more along the lines of not being exposed to it, to not have to endure it. The difference is subtle.
For instance, you would say "A person can't go without water for a week, or he'd die." A native speaker would not use do without in this sentence. Similarly, the sentence, "I could really do without all this noise" cannot use go without.
Of course, in sentences such as "You can go without me," the meaning is altogether different.
File this one under "crazy English." :)
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