It's not to do with peak loudness. Nor are the convertor chips in expensive interfaces necessarily any better than the chips in the less expensive ones, often they're the same. The biggest difference lies in the supporting circuitry.
As for loudness, humans perceive loudness primarily from how loud something is on average. Which means in audio terms we pay more attention to the RMS level than the peak level. Very short transients, such as the initial transient from a snare drum or banjo, can be incredibly loud, loud enough to be almost painful if sustained, yet be ignored by our ear/brain combination.
Lower actual volume, but sustained, tends to sound much louder. For example, a guitar distortion pedal or fuzz can sound incredibly loud compared to the "clean" sound when the pedal"s off. But when measured the distorted sound can be actually much quieter than the clean peaks. Guitarists sometimes find this out the hard way when they stomp on an overdrive, which sounds massively loud, only to have their solo disappear because the volume has actually dropped.
So making something sound louder is more to do with raising the RMS level compared to the peak volume than anything else. This is done by the, hopefully careful, use of compressors and limiters.
We also tend to perceive certain kinds of (slight) distortion as more musical than no distortion.
As for clipping, there are two kinds. One is the analogue kind, which can be emulated (though not always perfectly) in software, where the waveform starts to get compress with the extremes chopped off. Guitar amplifiers and distortion-based effects are the classic example. The other kind is where a digital circuit is overloaded and runs out of ones and zeros to represent the sound. At that point you get a very loud, very nasty noise with no musical content at all.
There are plenty of examples of music (and advertisements that are much louder than the TV programme) that's limited to the point where the waveform looks like a rectangular brick, with very little in the way of dynamics. It's actually quite hard to listen to for any length of time because it's tiring. The lack of dynamics can also give the impression it's quieter than it actually is once our hearing has shifted to accomodate the volume.
And as has been said, the only way to compare two different masters is to ensure they are listened to at the same volume. Until our hearing "shifts" we tend to prefer the louder track. Demonstrating that a more expensive system is "better" by raising it's volume slightly compared to a "not as good" system is an old hi-fi showroom sales trick.