What is the significance of "magic numbers" of atoms in nanoparticles?
As indicated in the comment above, "magic numbers" (i.e., atom counts with much higher stability) have been known in atomic clusters of many types for a long time.

Each magic number has a specific geometry. Some are non-spherical, so it's hard to make generalizations about surface area or lattice structure.

A very famous non-gold example is buckminsterfullerene ($\ce{C60}$). Multiple groups had seen the signature of 720 Daltons in mass spectra of soot for years before the official discovery. There are other magic numbers for $\ce{C_n}$ clusters.

In nanoparticles and nanoclusters, this is an active area of research, but the main difference is stability. The so-called magic-numbers are simply more stable. (Case in point, the thiol-termated gold nanoparticles were created by heating up the samples until only the most stable ones survived.)

While I'm not an expert, I suspect the increased stability likely means less reactivity, lower catalytic activity (i.e., the species is more stable, less reactive so less likely to activate).

I'll give a comparison with Hückel's 4n+2 $\pi$ electron rule. We know that benzene is aromatic and has increased stability relative to the 5-membered and 7-membered conjugated hydrocarbon rings. It's less reactive and thus requires specific reaction chemistry.