An onager isn't weight-activated. It uses torsional energy stored in twisted ropes. Operationally speaking, I think onagers would have offered the advantages of being compact; easily assembled/disassembled/transported; and you don't need a whole lot of precision in their construction. Downsides would be that your motive power depends on ropes, which can be affected by things like rain, rot, and so on; plus which you're somewhat limited in the size of your projectiles.
The catapults in the last LOTR movie were trebuchets. They use conservation of momentum -- put heavy(!) weight on one side of a pivot, and a sling on the other side. Primary advantage is that you can throw very large things a long way -- further than any other sort of catapult. Downside is that they're large, hard to disassemble/transport, and require a great deal of precision in their construction. (There's a way cool NOVA special about trebuchets.)
I think I've seen the Nova special. I've been a trebuchet fan, but I've been using Onagers in the Rome: Total War game I've been obsessed with, along with Scorpions, Ballistas, and repeating ballistas.
Rome's military supremacy was based on their engineering, I think I'm safe in saying.
Let me throw my two cents in...Rome fell primarily, because of 3 factors. The 1st was social: Families didn't want to send their "boys" off to war to defend Rome; and the soldiers rather fight for who paid them most, rather than their homeland.
The 2nd was political: Too many leaders were corrupt and interested in their own political gain/money/power instead of their countries.
The 3rd was economic: The barbarians (Huns, etc.) were disrupting trade in the empire...this led to less trade...etc. (I'm oversimplifying it b/c of time)
The Final Nail in the Coffin was Barbarian attacks. The barbarians had the will, not the might. The Romans had the might but not the will.
Sound familiar?
Just Maybe The LESSONS of History are worth something.