The Apocalypse is a good mission runner with its range bonus and amazing cap (with skills, it's over 9000). Fitting can be tight, but you can run beams or pulses w/ 3 lowslot mods or dual rep tank. The navy issue's larger grid and extra lowslot are also very nice if you like to have your cake and eat it too.
I am currently using this:
[Apocalypse Navy Issue, mission L4]
Large Armor Repairer II
Armor EM Hardener II
Armor EM Hardener II
Armor Thermic Hardener II
Armor Thermic Hardener II
Heat Sink II
Heat Sink II
Heat Sink II
Tracking Computer II, Optimal Range
Cap Recharger II
Cap Recharger II
Cap Recharger II
Mega Modulated Energy Beam I, Multifrequency L
Mega Modulated Energy Beam I, Multifrequency L
Mega Modulated Energy Beam I, Multifrequency L
Mega Modulated Energy Beam I, Multifrequency L
Mega Modulated Energy Beam I, Multifrequency L
Mega Modulated Energy Beam I, Multifrequency L
Mega Modulated Energy Beam I, Multifrequency L
Mega Modulated Energy Beam I, Multifrequency L
Large Auxiliary Nano Pump I
Large Capacitor Control Circuit I
Large Capacitor Control Circuit I
<Drones>
The same fit more or less works on the standard Apoc (minus a lowslot, obviously) if you switch to pulses and use a second tracking computer to get a little extra range. The Apoc's range definitely makes up for its lack of a damage bonus when fighting enemies that like to orbit at range (namely Sansha and Blood Raiders, who you want to be fighting with lasers anyway). The range bonus is especially attractive if you are unable to fit T2 pulses or don't want to use Scorch L. The laser cap bonus is nice if you use beams and the ship's very strong capacitor in general means you can fit interesting things instead of cap mods.
In response to your second question, people usually prefer battleships because they are a cost effective way to field the DPS you need to clear level 4 missions in a timely manner while also not dying. The Damnation is probably a poor choice for running missions since it doesn't pack much of a punch at all. It is meant to provide gang bonuses and give the FC a place to sit that doesn't make an attractive primary. If you want to run missions in a command ship, check out the field command versions since they have more DPS. They may, however, have difficulty delivering that damage to target due to range and speed issues.