It looks like possibly the mantra version has a higher absorption coefficient than the Eddy version? If you’re using the albedo version of the shaders then a higher absorption corresponds to a lower albedo and/or higher extinction. The absorption and extinction coefficients are scale-dependent too, which can be another source of differences.
The black body ramp used by the fire shader will use the physical black body colours (black to red to orange to yellow to white and finally blue for very high temperatures). One setting you might want to change is the white balance, as the default value for this is probably too low, so you might want to change it to 5000 or even higher, which might improve the range of colors. This will reduce the emission strength as a side effect so you’ll probably need to increase the emission coefficient to compensate.
The tricky part for fire shaders is getting your temperature to map onto the black body ramp in a nice way. The temperature coefficient in the fire shader is a simple scale for the temperature. Setting this to a value less than one will shift the temperatures towards the black/red side of the color ramp, values higher than one will shift the temperatures towards yellow/white.
A good way to fine-tune the colors is to use an E_Remap node on your temperature channel. Turn off the clamp option and set the ‘old max value’ and ‘new max value’ to the same value as the ‘max temperature’ in the fire shader (by default this is 5000). Then you can tweak the ramp to darken/brighten specific temperature ranges, while still keeping the full range of colors.