More watts are indeed louder. Every doubling of power increases SPL by 3 dB, until the speakers start compressing. With typical speakers in typical house sized rooms, average power levels are around a watt or so (to hit somewhere between 75 and 80 dB at the listening position - a fair bit more than twice as loud as normal conversation). More power would be louder, so at any given SPL into a particular speaker, the power has to be exactly, precisely, unmistakably the same. Ohm's law is, like, an actual physical law of our four dimensions of the multiverse, so a more powerful amp cannot, cannot, will never deliver a "different kind of" 2 watts than a less powerful amp. Of course once you push the amp into distortion, to the extent you can hear it over the speakers' distortion, which likely started 10 dB earlier, differing distortion patterns can indeed make things sound different.
And if you listen really loud, pushing your speakers into compression and distortion, a more powerful amp will indeed make that push harder and louder, and it's lack of distortion might very well be a boon. It's not what anybody would call high fidelity, but I'm pretty sure it's why some folks say you need more power for some speakers (like my former Magnepans).
You will of course hear folks say that more powerful amps sound different at low power. Not possible, just based on power. You also can find, with little effort, the audiophiles who believe that since average levels don't use more than a watt or two, lower powered (and usually class A topography) amps simply must sound better than high powered amps. Click your heels together three times and choose your preferred dementia.
Now, there actually are differences between amps that might make them sound different, especially when pushed hard, especially into difficult speakers. Output impedance could be high enough to change the frequency response of the amp into low speaker impedances. No self respecting audio idiot talks about output impedance, because they all have pulled their power ratings out of their flys and are shaking them at each other. And some amps measurably depart from accuracy (typically with a high frequency rolloff) and they may well be audibly different. But not because of their power. If it's what you like (personally, I am always seduced by a FR dip in the 2 to 4 KHz range, which is more likely to come from a speaker than an amp) it's fine to like it. I prefer, in part because it's what I grew up with, the grind of a mid-60s Fender guitar amp to that of a Marshall, even though many of my favorite guitar tones (Hendrix, Allman, Clapton, etc.) are Marshall sounds rather than Fender. (SRV is mostly that 60s Fender sound, though). One nice-ish thing about getting older is I am slightly more capable of listening past my biases and enjoying great non-Fender amps like an AC 15 or the handwired Marshall 1974. Gorgeous, really. But in a guitar amp, we are talking about preamps, power amps and speakers that are nowhere near flat (accurate) and are all typically driven into at least a little distortion (I've read that guitar players will identify an amp at 10 percent distortion as "clean" - that's way past horribly distorting levels for stereo or AVR reproduction . . . .)
Now, being loud or bossy or repetitive doesn't make either the power aficionados, or me, right. Reality is what it is. You can endeavor to understand both, and the net is your friend, ultimately. just expect to get dragged through a lot of horse apples on your way to actual understanding.
