This won´t help much but,
Can´t say that I´ve done a fair comparison between them, cause I never put Phoenix on every channel in a mix.
Just used Phoenix on separate stuff that needed a bit more balls.
Don´t use the Phoenix much anymore tho, just the heat on everything cause it´s so damn easy to setup and sounds good.
Heat can be both "tapey" and "tubey" depending on which way you set the drive knob, and I use the one that sounds best on the specific material.
Usually "tube" = clockwise, but it´s different from mix to mix.
From the HEAT manual
The Drive control introduces non-linear process- ing that emulates magnetic recording tape com- bined with harmonic information commonly found in analog gear. Turning the Drive control counter-clockwise emulates a tape based non- linear distortion (odd harmonic, starting with the third and fifth. As you increase the level on tape, or in this case the Drive control level, the harmonic content increases and higher fre- quency odd harmonics are added. A tape re- corder also contains record and reproduction equalizers that modify the harmonic content.
The result is a sound that fattens up the bottom and midrange of a recording and smooths high frequency transients as you increase the process. Turning the Drive control clockwise has a more aggressive effect, while adding an even har- monic series commonly found in triode tube cir- cuits to the odd harmonic series. This can give the effect of prized tube microphones and other tube based analog gear.
Note that the harmonic structure changes with signal level and the amount of processing being applied, just like it does in a tube and the analog world.