In general, use active boxes for instruments with passive pickups, and passive boxes for instruments with line-level outputs like keyboards and acoustic guitars with battery-powered preamps. Passive basses tend to sound good through either one.
Like gavin said, many phantom-powered DIs sound bad when starved for voltage.
Fletcher, I'm interested- the Axmax says it is DC powered. Does it have a power supply or does DC refer to phantom-power? Where can I find specs? I couldn't find any additional info online. A wall-powered DI that can handle line inputs for $75 would be a great deal.
For inexpensive passives, I've alway had good luck with Whirlwind Directors.