What's the point in a freewheel/cassette with incrementally sized rings? Pictured is a freewheel with 15, 16, 17, 18, 19, and 20 tooth sprockets. What I don't understand is why such small increments are worthwhile. Surely the point of having gears is the variance that they provide? Please enlighten me.
The point is efficiency.

Cyclists actually have quite a narrow optimal power band. Most of us can bang away at a cadence of 50 rpm, up to about 90 rpm. Some of us pedal faster - 90 to 120 rpm. At those lower cadences (50) it feels like we're producing lots of power and at the high cadences it feels like we're just breezing along. But that is confusing torque with power. The optimal power band is where we produce most power most efficiently.

How narrow is that power band? Take me as an example: 50+ years of road cycling, leg muscles well trained to buzz along at 100 to 120 rpm. My optimal power band is 105 to 115 rpm - on a flat ride I can do that for hours. If my cadence is dropping on a rise or into a headwind and I'm starting to grind, I know I'm down around 100. On a hill, after I run out of gears, if I can maintain a cadence of 90 or above I'm happy. But it's not as efficient.

So there are three factors to consider: the rider, the terrain, and the wind.

Terrain. On a flat road, close spacing gives us scope to choose the optimum gear. On a hilly road, we might want wider spacing at the bottom of the range to help with climbing. For MTB, AFAIK such a cluster would be unsuitable, but I'd welcome input from an MTB rider.

Wind. On a circuit, wind is like hills. Except that we always seem to have a headwind. Whatever. In wind a cluster with wider spacing helps.

The Rider. You. Where is your power band, and how wide is it? With training it will change. You'll hear many riders say that they "top out", meaning they couldn't pedal any faster (look at the tables below and see what that means). More factors come into it now: training and pedals. Pedaling fast takes practice, not just to pedal, but to do it smoothly so you don't bounce uncomfortably. And you have to be "clipped in" - no flat pedals.

But I digress. A general purpose 10-speed road cluster for a strong rider might be


  11, 12, 13, 14, 15, 17, 19, 21, 23, 27


The difference between each gear at the top end (11 to 15 teeth) is between 6% and 9%. At the bottom end, the difference is about 10%, except for the "granny gear". Taking our narrow power band into consideration, and assuming we have 39x53 front chain rings, that means we'll be able to find a comfortable gear at a wide range of speeds.

On the 39T chain ring, at 90 rpm (as an example; it's the ratios that matter, not the exact cadence):


  19 kph on the 23T sprocket
  21 kph on the 21T
  23 kph on the 19T
  25.8 kph on the 17T
  27.5 kph on the 16T
  29.3 kph on the 15T
  31.4 kph on the 14T
  34 kph on the 13T
  37 kph on the 12T
  40 kph on the 11T


And still on the 39T chain ring, at 110 rpm:


  23 kph on the 23T sprocket
  25.5 kph on the 21T
  28 kph on the 19T
  31.6 kph on the 17T
  33.5 kph on the 16T
  35.8 kph on the 15T
  38.3 kph on the 14T
  41.3 kph on the 13T
  45 kph on the 12T
  49 kph on the 11T


So, if you were riding at 30 kph, you could choose from the 15T cog, the 16T or the 17T. One of them will (in my case the 16T) be closest to your optimum cadence.

Switching up to the 53T chain ring, at 90 rpm:


  26 kph on the 23T sprocket
  28 kph on the 21T
  31 kph on the 19T
  35 kph on the 17T
  37 kph on the 16T
  40 kph on the 15T
  43 kph on the 14T
  46 kph on the 13T
  50 kph on the 12T
  54 kph on the 11T


And still on the 53T chain ring, at 110 rpm:


  32 kph on the 23T sprocket
  35 kph on the 21T
  38 kph on the 19T
  43 kph on the 17T
  46 kph on the 16T
  49 kph on the 15T
  52 kph on the 14T
  56 kph on the 13T
  61 kph on the 12T
  66 kph on the 11T


So now, considering that we have both chain rings available, you can see that at 30 kph you have 2 more possible gears. So depending on the terrain ahead, you would choose the big or small ring and the appropriate back cog.

For even stronger riders, the bottom gears are only 21 and 23, or maybe as low as 25.

Hope that helps!