Generally, fan translations are quite literal - often overly literal to the point of needing a great deal of footnotes and including untranslated Japanese terms in the English sentences. They tend to be more prone to errors as well, often hard to detect errors because a casual glance at the dictionary can give the reader the same misleading gloss for the same unfamiliar word giving the same misunderstanding. The tone of dialogue is sometimes hard to understand as well, often being translated into very similar English grammar without regard to what the natural English is for the situation.
Generally, publisher translations are quite liberal, often rewriting difficult parts for smoother reading (i.e., replacing untranslatable puns with another kind of joke, replacing a reference to a holiday unknown in the west with one that is known, changing the date references to make them mean 'present day' in the year of publication, just to name a few.)
Which one you want is really a matter of what you're trying to achieve and what you prefer.