It's not just whether or not it can handle power or what the gas mileage is. Something important to consider is how efficient the transmission is. This determines whether or not the car is actually drivable.
In powerful cars like a Corvette you can get away with as few as 3 gears and it still performs great. In something with 100HP peak and a little tiny 1.5L engine, efficiency becomes a big deal. If it only has 100HP, I want all it going to the wheels and I want access to it at all times.
In North America it's hard to make a test that shows efficiency differences since it seems to come down to 0.1 second differences. Differences small enough to be random error. Luckily our friends in the UK build ridiculously weak cars where the differences are much greater and easier to measure.
UK Honda Fit
ES model:
0-60 with 6MT = 11.5 sec
0-60 with CVT = 12.8 sec (11% slower)
The manual also has a slightly higher top speed of 113mph vs 109mph. Clearly something weird is going on here. Even when the CVT is able to hold the engine at maximum power the entire time, it still has less power going to the wheels due to energy loss in the transmission. 11% is not a big deal when you're driving a car with a V6, but you'll definitely feel that 11% in something like a Honda Fit or a CRZ or Prius when you're trying to merge onto an interstate highway or when you try passing something. If you want all of that power you paid good money for, then you'll want a manual transmission
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I'm really fascinated by the UK site since they post performance numbers for their vehicles. The one for the CRV is interesting because it has premium model diesel as well.
UK Honda CRV
0-60 petrol manual = 10.2s, top speed 118mph
0-60 petrol automa = 12.2s, top speed 110mph
0-60 diesel manual = 9.6s, top speed 118mph
0-60 diesel automa = 10.6s, top speed 116mph
So pretty much any way you slice it, conventional automatic and CVT both lag behind the manual when it comes to power delivery.