Apples and oranges, really.
The ES9601K comes in a tiny QFN package and probably runs on +3.3 V (or +5 V), or at least that's what I'd infer from 2 Vrms with an internal charge pump negative supply. The product page says nothing about it being Class G/H (like the Maxim ones would be), even though this would probably be a good idea given the limited thermal capabilities of such a small package. This would be the kind of chip to use in a portable player or smartphone.
The TPA6120A2 originally derives from a DSL line driver. It operates on a traditional bipolar power supply from +/-5 V up to +/-15 V, delivering anywhere from ~1.4 to 8.5 Vrms (so up to 18 times the power into high-impedance headphones). As power dissipation can be over a watt, an exposed power pad on the bottom of the package is used to transfer the heat away from the chip (if you can't solder that, you can always turn the chip upside down and put a heatsink on top). Minimum advisable load impedance for this guy is probably 64 ohms, performance tends to suffer below that. Applications often use 10 ohms of output impedance to get stability in check.
Note that both basically are drivers only. In addition, them directly following a DAC as suggested in documentation will require said DAC to be very well implemented, with excellent real-life performance, if you want to avoid excessive output noise. The 117 dB of the ES9601K may still be somewhat realistic, but getting close to the 128 dB(A) of the TPA6120A2 is very likely to require an analog volume control and opamp buffer / gain stage (note that gain structure is likely to be different between solutions involving a volume pot or a PGA).