Not all synthetic materials are created equal. The most common ones in use today (Under Armor, Nike Dri-Fit, etc) are pretty basic from a technological standpoint (because they are CHEAP to manufacture). There are other options, but how much 'better' they are depends on a number of factors.
Synthetic materials work by using hollow fibers that 'wick' (through capillary action) moisture off from your skin. This is a fast process, once liquid moisture is present, but it has the down side of creating lots of areas for bacterial growth. In old polypropylene base-layers it was nigh-impossible to clean all of the moisture channels, resulting in their notorious permanent stink. Modern fibers are able to be more effectively cleaned, but the smell will linger until washed. This is addressed by some manufactures by adding a small amount of silver to act as an antimicrobial agent. The down side is increased cost and some potential environmental risks (in manufacturing).
Natural materials transport moisture along the outside of their fibers. They are also more loosely packed, so moist vapor is able to pass trough, not just liquid. The end result is a slower 'wick' but without the need for liquid to build up on the skin. Many natural fibers (like wool and bamboo) have natural anti-microbial properties which allow multiple uses without the stink that synthetics develop. The down side is that detergents bond to the surface of these fibers and can reduce their performance over time. There are special tech-wash products out there for cleaning/restoring such high-end base layers.
A note on cotton
Cotton loves to absorb moisture, which is why it makes a wonderful towel, but a terrible base-layer. Cotton socks/shirts/underwear are the utter bottom tier option for performance stuff. If you've ever removed your white cotton sock and gazed at pruney wet toes, you have seen it at work. Avoid cotton for anything where sweating may occur, as it can/will lead to blisters and general discomfort. Linen is better than cotton in nearly every respect (for warm weather casual wear).
Given that both material choices come in multiple weights, you can dismiss the "one is cooler/hotter than the other" argument; just buy the appropriate piece for your needs. What it comes down to is a question of environmental concern (synthetics are worse for the environment than naturals, because they have to be manufactured) versus performance needs (synthetics wick faster, insulate better than most natural fibers, etc) and comfort (natural fibers tend to 'feel' better for most people, i.e. not 'sticky').
In summation here is a small table to pull from.
- Dry faster
- Easier care
- Environmental damage
- Require liquid sweat to build up
- Feel 'sticky' when wet
- Transport vapor
- Natural anti-stink
- Long lasting
- Higher maintenance
- Specialty shops/online only
[Good brands to look at for Synthetics: ExOfficio, Patagonia, Mountain Hardwear]
[Good brands to look at for wool: Ibex, Smart wool, icebreaker]