There has been (which was starting when this question was written but has become even more so in the years since) a trend in PC hardware design away from 3.3V and 5V and towards 12V. The modern low voltage (often less than 1V) electronics can only practically be supplied by point of load converters and it's more efficient to run those power converters off a 12V input than a 5V input.
Unfortunately the ATX power connector was designed in the days when 3.3V and 5V dominated. The 20 pin ATX power connector only has one 12V pin, the 24 pin connector has two.
The CPU has it's own 12V connector so we don't have to worry about that but we do have to worry about general motherboard loads and any expansion cards. IIRC a PCIe graphics card is allowed to draw up to 75W from the motherboard (beyond that they use a supplementry connnection to the power supply).
Add a PCIe graphics card to the various loads on the motherboard and you are up to arround 100W or so.
A mini fit JR is supposed to be good for 9A. The newer mini fit HCS (which is pin compatible but higher rated and is reccommended by the ATX12V 2.0 standard) is rated a little bit higher. 9A at 12V works out to 108W. Unfortunately however PC vendors don't tend to use genuine molex connectors. They use cheap copies which often can't handle the current.
So in summary.
- In a machine without a graphics card a 20 pin PSU will almost certainly be fine.
- In a machine with a single graphics card it's borderline, a 20 pin PSU should be ok if all the parts are high quality and in good condition but there is a real risk of burnout.
- In a machine with SLI/Crossfire using a 20 pin PSU would be a very dumb idea.
The adaptors you mention aren't strictly needed, you can just plug the connector straight in. The advantage of using an adaptor is that if you do burn out a pin it's a pin on an adaptor rather than a pin on the motherboard which is easier to deal with.