what is better, an aquaclear or sponge filter?

I just love my Aqua Clear 500/110.

on a 55g my first choice would be a canister filter for heavilly planted tanks, but my second choice and least expensive, is the aquaclear. I'd go with an aquaclear. Its easy to maintain and you can vary the filter material as needed. I used 3 sponges an overstocked stocked 25g and everything came thru fine (holding for a few months until I setup my bigger tank).
Now I'm using the ac200 and ac150 on my 2 x 25g tanks respectively,and its just peachy.
I'd go for the biggest ac available, and later on you can add a sponge filter too or maybe another ac if you want really high filtration in relation to what your keeping in the tank.
I'm using co2 in my 25g betta tank, so what I did was buy a thin kitchen pot scrubbing sponge, stuck it into the outpour so it hangs down,to reduce splash (and noise too) and to keep the co2 in the water. It works like a charm. Furthermore, the ac's are very reliable and I've found, they will startup after a power failure, if there's enough water in the reservoir. Which is usually the case. Easy to prime by just pouring water into it. Whereas my eheim canister can be finicky and difficult to get primed.

There........I think I said it all. Phew.....

not diatom

No, you do not want a diatom filter, that is for special purpose only, not general filtration needs.

If you want really cheap and have an airpump, you can get a hydroV sponge and have good biofiltration, not great mechanical filtration, but you can gravel vac to keep floor clean. I have one also hooked to an airpump that comes on if the power goes off, so in a powerfailure my tank still has filtration going.

Next is the AC line, very good mechanical filtration, not as great biofiltration I think, as the flow rate is so much higher through the sponges. Best bang for the buck, probably. Get advice on sizing, I don't know but have been amazed at the flow from rather small filters, it might be easy to get a filter too fast for the tank. These filters may overflow onto the floor if too dirty, I've heard, and may need a push to get restarted after power failure.

Next in line, IMO, is Penguin 330 (now 350?) with dual biowheels and cartridges and media baskets. You can get replacements for the bought cartridges that are a clamshell type frame to fill with cut to fit pads, I like mine. Good mechanical filtration with the frames or cartridges, good biofiltration with the biowheels. Adequate for a 55 IMO. Restarts after power failures, never leaks onto the floor. Not for use with a planted tank which has CO2 injected as it loses CO2.

All of those are good and cheap for the not-planted tank. Was there some indication that you were doing a planted tank? I didn't see any.

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rates for sponges

I would think that using an air pump on the Hydro sponge would be bettter, for a gentle flow is much better for biofiltration, while a blast will give you good mechanical filtration but the dwell time is too short for biofiltration. Although, really, I think the 330 alone will be enough. I used a sponge with the 330, but I had 23 baby discus in there getting fed 6x/day!