The HeadRoom Total Bithead and ProJect Head Box II are two very different amps. The obvious big difference is that the Bithead is designed to be portable, whereas the Head Box is definitely not. I presume you would be using the amp for home/office listening?
The Bithead also has a DAC (Digital to Analogue Converter), which is great for computer listening as you bypass the nasty headphone output present on most PC's and laptops.
Have you considered an
iBasso D2 headphone amp & DAC? Functionality is very similar to the Bit Head. I haven't tried both side by side, so can't comment on performance.