Batteries have no technical or safety advantage over a wall wart. Well, that was easy.
In fact, other than portability, batteries are worse in almost every technical aspect, such as non-constant output impedance, an ESR which increases with time and load, cost, and the biggie, an output voltage which by definition is not constant over zapper-type time periods, such as 1 to a few hours. Add in cost and environmental issues (producing a battery is an environmental nightmare), and batteries just suck. Lithium cells are better than most, if they don't burst into flames.
As for safety, there is nothing in the history of electronics more reliable and safe than a wall wart. The world market consumes several BILLION each year, and the failure rate is so small that it is very hard to find data. It is more likely that you will be electrocuted making a phone call during a rain storm, something for which we actually have data.
The only thing I can think of to watch out for is that most zapper circuits are poorly designed from a power management point of view. A well designed circuit (as opposed to most zapper designs) has a small capacitor located near the power pins on the chips, to perform a function called "decoupling". The capacitor reduces the apparent output impedance of the power source, something which keeps the chip happy. Fresh batteries have a pretty low output impedance, and usually are connected to the circuit with relatively short wires. Because of this, a 555 or 4069 type zapper circuit functions ok without a decoupling capacitor, although it would be measureably better with one. The original Clark circuit did not have one, but that's another story. Anyway, a wall wart usually has 3 to 6 *feet* of wire on the output, a significant increase over the typical battery connection. A zapper circuit based on the 555 probably won't care (because those boys at Signetics were very good at their jobs), but some zappers designed with batteries in mind might. One designed from the ground up for AC power should have no problem. The UZ has decoupling caps.
As for UZ... I've always been disappointed in the text on their site. Things like "100% positive pulsation" and "constant wave" sound techy, but actually say nothing clearly. All zappers have a unipolar output. An AC powered unit will have a more stable output amplitude over time, but there is nothing in Clark's
"research", such as an acceptable output voltage range, to indicate that small variations are a problem. So I wonder why someone with an above average command of electronics would write such junk. I wonder...