Ahh, how very many "which one" threads I answer...
Okay, here's my thoughts: Capcom got smart this time and made it a battle of taste in games than "OooOOOooh - dat wun wooks cooler!!" SO... generally speaking, Gregar is for those who desire flat-out power, and Falzar is for those who plan strategically.
Gregar is based towards power-players. The Crosses like Heat, Elec, and Erase all cover one row but can do considerable damage. The LinkNavis themselves are more often prone to element weakness found in many viruses, and their charge shots don't cover much - yet they pack a serious punch.
Spoiler (click to toggle)
The BigHook and ColForce GigaChips are also based on power - BigHook does strong, undodgeable damage (w/o Invis) and ColForce just blasts the tar out of people.
Falzar is bent toward a more strategy note, seeing as the Ground and Tengu Crosses both have the most bold strategic appeal - Ground's charge shot dives under enemy attacks and counters with its own; Tengu pulls enemies in for a WindRack hit. The linknavis here are less prone to element defeat and their attacks may not be the strongest, but they can be combined with others to become devastating.
Spoiler (click to toggle)
This is Falzar's ultimate strategic key - the HubBatc GigaChip. Tired of a cramped NaviCust? Fear no more! This chip maxes out your buster and custom screen, and gives you undershirt, shield, flotshoe, and airshoes to boot! Nifty, huh?
Regarding last bosses:
Spoiler (click to toggle)
Gregar is quick in the sense that he keeps the attacks coming, but his main moves leave him open for counter. Falzar avoids counterattack opportunities, but is slower on the attack-dishing. IMO, both bosses have crosses that own them and are easy once their patterns are known. Their SP forms should be tougher, though.