I've compared those two on 2 meters. Both antennas were homebrewed
based on W4RNLs' info on his web page.

The moxon has deeper side nulls, but they are at different angles than
those of the quad. The front-to-side nulls on the quad are pretty much
at 90 degrees, while the moxon's nulls were more close to 120 degrees.
I can't quantify it beyond S-meter readings on my VX-5R, but the moxon
nulls were deeper by 2 or 3 S units on this particular VX-5R. Your
mileage may vary.

Directly to the rear, the two antennas were about the same.

I couldn't determine any gain differences, nor measure a 3db beamwidth,
but the beamwidths were similar for horizontal polarization, to about 60
degrees either side of straight ahead. Then the yagi started dropping
off. The moxon dropped off more slowly until well past 90 degrees.

Vertical polarization is different, but you probably are not planning on
that for 20 meters. Vertically polarized, the moxon had a nice deep
null directly to the rear, while the pattern on the quad was more or
less unchanged from its pattern horizontally polarized. The moxon's
vertical beamwith was also quite broad.

In sort, what I saw playing around with the antenna's was very similar
to Cebik's modeling predictions.

All of my tinkering was done listening to local repeaters, with the
antennas at about 15 feet height.

For 20 meters, it will probably be a matter of which is easiest to build
and support. A monoband loaded yagi might be easier to homebrew for 20
meters.

73,
Keith, W4KAZ