Water is the biggest contributor to the greenhouse effect (~60% of the total), but that's because it's the most abundant by far (wikipedia indicates about 0.25% globally [0],
by mass, which translates to something like half a percent by volume; meanwhile CO2 is 0.04% and CH4 is 0.0002% by volume). And as others mentioned, it's mostly not human-generated, and it stays in the air for a much shorter period of time (on the order of days instead of years or centuries because of a localized phenomenon called, um, rain).
Methane, on the other hand, is much more than 2x as potent as CO2 (estimates range from 21x to 40x the warming effect over the span of 100 years, but most are centered around 25x, when taking into consideration that that methane's atmospheric lifetime is only ~12 years).
You could argue that there's a 3x multiplier in the comparable difference in weight, but you've still got an 8x multiplier. Even if we assume that CO2 and H2O are comparable in terms of warming potential by volume (it's hard to measure H2O for various reasons), there's still a 3x multiplier over that 100-year period, compared to 1x CO2 and 2x H2O.
Why would some molecules be more potent than others? It's a matter of the infrared wavelengths they absorb, but in particular how they cover the spectrum relative to other atmospheric gases. CO2 absorbs strongly in parts of the spectrum that H2O absorbs more weakly, and CH4 absorbs strongly in parts that aren't covered by either CO2 or H2O [2].
[0] https://en.wikipedia.org/wiki/Atmosphere_of_Earth#Compositio...
[1] https://www.epa.gov/ghgemissions/overview-greenhouse-gases#C... - "Pound for pound, the comparative impact of CH4 is 25 times greater than CO2 over a 100-year period."
[2] https://www.acs.org/content/acs/en/climatescience/greenhouse... which depicts the absorption spectra for various gases on the right: https://www.acs.org/content/acs/en/climatescience/greenhouse...
