The deeper the rim, the more aerodynamic the wheel will be and the more efficiently and faster you'll be able to cut through the wind. An 88mm deep rim is more aerodynamic than a 38mm rim, and will be faster in situations where aerodynamics are very important, like long fast solo riding and triathlons. Also, once up to speed, the 88mm rims maintain that speed with less work which means that you save a lot of energy on long distance rides.
Why would anyone choose 38mm rims, if 88mm rims are more aerodynamic? The more a wheel becomes aerodynamic for going forward, the more it can be negatively affected by cross-winds, or winds coming from the side. Let's imagine you are riding with an 88mm deep front and rear rim. You will be able to go very fast because of the aerodynamics of deep wheels, but if a gust of wind comes from the side, that wind will push sideways against the large rims and push the whole bike sideways. This isn't such a big deal if you're riding alone, such as in a triathlon, but if you are riding in a group of cyclists, this sideways push could cause an impact with other riders, or at the very least it would require extra concentration from you.
What's the solution? A wheel with a 'shallower' rim depth, like a 38mm, will be less affected by crosswinds, but will also have less aerodynamic advantage.
If that wasn't enough, you must also consider weight. A 38mm deep wheel will be lighter than an 88mm deep one, because less carbon is used. A lighter wheel will accelerate faster and will be much easier to ride up hills. This means that if you are sprinting and hill climbing a lot, a shallower rim, like a 38mm will be best.
Why do I see some riders with different front and rear wheels? Due to the fact that the front wheel can turn for steering, its more susceptible to cross-wind interference. The rear wheel is fixed in a straight line with the bike, so if a cross-wind hits it, the impact on the bike won't be as much. This means that you can use a deeper rim on your rear wheel without impacting handling as much as that same rim would affect the front wheel. Wheelsets such as the 38-50 to take advantage of this fact. This wheelset uses a 38mm front rim, and a 50mm rear rim. This way you can get more aerodynamic advantage without sacrificing as much stability in windy conditions.
How do I choose? This is the most difficult choice for riders looking to purchase new carbon wheels. You must think about the situations you ride in. If you ride lots of hills you will want to choose a 38mm wheelset, if you are an all around rider choose a 50mm wheelset. If you are looking for a a wheelset that may be half way between; a great all around wheelset, but with some added benefits for hill climbing, choose a 38mm/50mm combo. If you are an all around rider looking for more aerodynamic advantage or a solo rider/triathlete, choose a 60mm/88mm combo wheelset.