There's usually no relationship between SPL and FR (unless it's really, really, really messed up or you're kind of making some kind of analysis into some bizarre clipping / distortion mode when pushing it past what it can handle). If the FR changes at different SPL, you have something highly nonlinear on your hands.
Speaker amps don't produce vibrations and sound pressure levels, so that makes no sense. They're electronic amplifiers that can push certain power levels into speakers, which are the things making the sounds.
The loudness you get (in SPL) depends on how much power is delivered to the headphones and then the headphone sensitivity. The more power you deliver, the louder they get. The SPL can be directly calculated from the sensitivity and the power input level. For HE-6, that is literally a spec of 83.6 dB SPL given 1 mW input power. Supposing that figure is exact, you'd expect 93.6 dB SPL given 10 mW input power and 80.6 dB SPL give 0.5 mW input power, for example. For speakers you also need to figure out the distance from the listener to the speaker, but for headphones the distance should be (more or less) the same for every wearer so that's not really a consideration.
The amount of power delivered to the headphones depends on the capability of the amplifier and then some other factors. An amplifier is limited by the physical (electrical) parameters of the design and components, so different amplifiers will have different maximum power ratings of what it can handle cleanly before starting to clip. For a given amplifier, the maximum amount of power that can be delivered depends on the impedance of the load. Speaker amplifiers generally have higher maximum power ratings because—oversimplifying and glossing over transducer design for a bit—you need more power generally to fill a room than to fill the space between some headphones and your head. Of course. A lot of headphone amplifiers aren't capable of a clean 1 Watt or more into 50 ohms or so, while pretty much every speaker amplifier is.
Supposing we're talking about amplifiers operating within what they can actually handle, the actual amount of power delivered depends on the signal level sent to the amp (from the DAC or preamp), the amp gain, and the volume control. The speaker gain boosts up the signal level by some multiplicative factor greater than unity, while the volume control brings it back down if set to anything less than maximum. Generally, speaker amps tend to have higher gains because they more often need to be operating at higher volume levels. (the gain doesn't have anything to do with whether or not the device can handle whatever output level results from all the above though.)
Many headphone amps may not really be designed for the kind of power levels you need for listening to the HE-6 loudly, but some are. But as for why anybody likes anything, it's a combination of many factor, many of them having nothing to do with the electronics and the sound produced.
