The only conceivable advantage of the greater crystal of phoenix ash threat is that since the continuous damage component is, on average, larger, if the target attempts to cast a spell or perform other activities requiring concentration, the Concentration check is large somewhat larger (at least, 50% of the time—though 16.67% of the time, the greater crystal of fire assault is better, and the rest of the time they’re even). But 5 damage results in a DC 12 Concentration check—something no one who cares about Concentration should be failing under any circumstances. A greater augment crystal needs a +3 weapon to attach to, so someone who has either crystal has a minimum of 24k gp spent on getting it—and this is not going to be anyone’s top priority. Buying it before 10th level is extremely unlikely—which means level-appropriate targets have a +13 to Concentration from ranks alone. They literally cannot fail the concentration check from either crystal. But this is, theoretically, the one thing the larger damage over time has over getting a portion of that damage upfront.
Basically, whoever wrote the crystal of phoenix ash threat didn’t understand the system very well, and was not actively comparing to even to the other options offered in the same section of the book, to say nothing of options beyond weapon augment crystals. It serves no purpose but as a trap for players. If it showed up in randomly-rolled loot, it’s easily worth selling for 3,000 gp at the first opportunity. Before then, if you have a +3 weapon—which you probably shouldn’t, ideally, but supposing you do—and no other weapon augment crystal, you would probably use it until you came across someone who was willing to take it off your hands.