It's a good question, and one that comes up often. Here's my little essay on the topic. It's just one person's opinion and experience of course, and we all tend to like what we're used to.

For tone, my preference is for high 4th in C tuning (on a small uke) and low 4th in D tuning (on a big uke). Of course, it does depend on the uke, but that puts both tunings in about the same pitch range. I use low4th D almost exclusively, but I have a soprano with a high 4th and I understand the appeal. If I were sure I'd live to be 110, I'd work more with it. (I also have a Venezuelan cuatro, where the first string is an octave low. It's just weird.)

There are pros and cons for everything. In fact, the pros of one tend to be the cons of the other, and vice versa. It's not that one's better, they're just different.

High 4th (also known as re-entrant tuning)

PROS:
High, close chord voicings- which are attractive, and most people think of as characteristically "uke"-y. Also, having a high note at the top and bottom of the strum gives a different rhythmic effect, which is, well, uke-y.

Many nifty solo and melodic effects can be had by using that high 4th string as a melody note: cross-picking, banjo style, and what they called campanella in the 17th century (when the standard guitar tunings were re-entrant, like the uke). 5-string banjo players call it "melodic style".

Keeps all the strings in the range- that sounds best on a small uke. A small instrument seems to have a narrower range of best-sounding pitch than a big one, especially if you don't want wound strings. I'm pretty sure this is how re-entrant tuning got started in the first place.

CONS:
Narrow range- your open strings only span a major sixth - a problem if you want to play melodies with a wide range, especially in a variety of keys. On a 12-fret C tuned uke, you've got less than two octaves, c'-a". 15 frets give you the high c". It's not a problem if you use your uke mainly for accompanying, as most people do.

Stinky chord voicings - a lot of fancy chords are going to include semitone dissonances that sound better as sevenths or ninths. Not so bad in a group, but sometimes there just isn't a solution playing solo and you have to change a harmony. Also, many plain chords have unisons in them, which can sound a little bare and makes fine tuning crucial.

Low 4th Tuning:

PROS:
Wider range- takes you a fourth lower, of course, and gives a 15-fret instrument two octaves and a fourth, a-d" or g-c". 17 frets on a D-tuned tenor gives you a-e", a quite respectable range for classical music or jazz, more than some wind instruments and most of the violin range that was used until the 19th century. This gives you access to fiddle tunes, Baroque music, and more flexibility of keys. The low 4th tuning is generally more flexible and able to do more things, mostly because of the added range.

More spread-out chord voicings- makes more dissonant chords sound better, usually, and gives you more variety of voicings. I think you have to think about voicing a little more, it's more automatic on the re-entrant tuning.

CONS:
Harder to get a good even sound- especially on a small uke. In fact, it hardly works on a soprano size, even with a wound string. (And to my ears, a wound string violates the integrity of a uke - but that's my prejudice; some people dislike the sound of a low 4ths, I dislike a wound string on any uke.) I think low 4ths are for tenors and baritones, maybe some concerts. (I know Don Ho and Chalmers Doane both play concert-sized ukes with wound low 4ths. They're great players, playing high-end ukes, and I still sometimes hear things that I think would sound better on a bigger uke with an unwound 4th.) Trying to tune a uke too low for its size - no matter what string you use - is going to give you that dead, floppy sound, and a fourth string that booms out, won't blend, and won't play in tune. Listen to your uke! Tune it to love it!

Some people are always going to tell you you've got guitar envy. If what you love about the uke is that re-entrant sound, then by golly that's what you should play. But in some sense any uke is a kind of soprano guitar. So what?

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John Kavanagh
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