Some headphones have higher than 600 ohms.
Loudness is determined by output level, headphone impedance, and headphone sensitivity—usually given as a figure like 95 dB SPL / 1 mW power input. For sensitivity, some headphones go below 90 dB SPL / 1 mW, while some IEMs are more like 120 dB SPL / 1 mW. This is a huge span, so focusing on impedance alone is ignoring a very important factor. Some 600 ohms headphones are less sensitive than the Beyerdynamics or whatever you may be thinking about. Because of the low sensitivity, the ~50 ohms HE-6 needs a higher voltage than 600 ohms Beyerdynamics to reach equivalent loudness. Because of the low impedance and P = V^2 / Z (power equals voltage squared divided by impedance), that means the HE-6 needs a whole lot more power than those Beyerdynamics, for example. Higher-impedance headphones are easier to drive in general , except that they need a higher voltage equivalent power levels (again, see P = V^2 / Z). Amplifiers will usually have more problems with lower-impedance headphones, because more current is required and because of potential output impedance issues.
Non-idealities in amplifiers (intended by the designer or not) mean that there is some difference in signal produced between different amplifiers, particularly at different operating conditions, so sound quality can be affected. Most people will point to different amps sounding different, and this is the big draw behind having different amplifiers—though maybe in reality, differences are less than people think. In controlled listening environments, fair comparisons, comparisons with different amps under bedsheets, the ability for people to correctly identify different amps is diminished (though not to zero of course, for certain combinations)...
Some amps look nice; some people like having volume knobs to turn. There are reasons other than sound quality to get an amp, though most people try to talk about sound quality.
A common situation these days is to have people using but a fraction of the output power of an amplifier (e.g. 10 mW out of a possible 500 mW @ 50 ohms), and they say they don't like the sound they get from their headphones. Particularly for planar magnetics, the common response seems to be to get a different, even more powerful amplifier (e.g. capable of 2000 mW @ 50 ohms), with the claim that you need more power. That's bullocks. You can try something different, but unless you're running your current amp out of its comfort zone, the sound is not a result of lack of power. It's more likely that someone just doesn't like a certain headphone's sound, which is a-ok. There are other headphones out in the world.
So to answer the question in the title, higher output power is needed if you want to listen louder. Headphone sensitivity and impedance determine how loud they will play for a given setting on an amplifier. If you max an amp out and want more volume, or if the amp is distorting at higher levels, you want more power. Otherwise, no.
In my opinion, the race for high output power headphone amps is mostly a marketing sham. People generally are not using the extra power. It is very convenient for amp manufacturers that there is a lack of understanding among consumers about what electric power means. An amp capable of higher output power may or may not sound subjectively more "powerful", with more "authority", and so on with the buzzwords.
