Besides allowing more air to pass under; what are the benefits or downsides of a cat hull with a center pod (like an eliminator or DCB) opposed to the side pods only (such as a Skater or Spectre hull)?

Not sure of any real advantage to a center pod. The twin sponson hulls allow more air to compress, increasing the speed and are better in rough water since they give more wave clearance. Usually you only see the pod in smaller lake boats.

The center pod helps eliminate porpoising below 85 or 90mph on a larger Cat as well as a great overall stable ride.

Over 90mph packing air the center pod on my 37' rig isn't even touching the water until a hard turn or rough water.
Rough water is where the traditional tunnel hulls will ride much much better,over 3' surf my boat will pound you hard as hell .
In the 3' + water a 36' Skater feels like a magic carpet. Twice the hull efficiency with a true tunnel vr a center pod.
We make up for the lack of efficiency by running ultra high HP and a correct set up, drive ratio ,props etc...

some center pod cats are single engine boats and it is there to accommodate the engine and drive as well.

I have a 25 foot center pod cat. I really like it in smooth to to 2 1/2 - 3 ft waves, accelerates hard, good low speed handling, little or no porpoise, great top end.

Can't really run over 30 mph in over 3 ft waves. Just when you think everything is fine, you can hit a rogue wave, sky the front, trip and stuff. If only one sponson gets stuffed, it can spin out. Just have to be real carefull in the rough.

I had a small Skater before (21), loved the rough water ability, hated the porpoise and low speed handling. It wallowed like it was full of water at under 45 mph and porpoised right at the speed I like to cruise, 55-70 mph, drove me nuts.

What hull is better depends on the water you like to run in.

I understand the center sponson has something to do with slow speed turning too, but I could never figure out the physics myself.

This quote is 100% on the money. I have owned both and l prefer true cats at high speed and rough water. On smoother water the modified tunnels (pod) are really nice.

the center pod also helps keeps the center engine in a triple engine outboard application from shearing off gear cases

That's strange. We used to run the smaller tunnels (or modified V, or Mod-VP) with a center pod (STV) and the pod was actually lower than the outer sponsons - hence the name, modified V. Be interesting to hear the different thinking behind the different designs.

Mod VP's like the STV's run with just the pod in the water and with the sponsons in the wind. totally different from big cat dynamics, some of those little things really hang it out there