Simple, if you could only have one or the other, which is better for an Antigua-line gun - Damage Increase or Attack Speed Increase?
What bonus does Antigua line benefit most from?
First, for unique variants on just about every weapon, you prefer ASI or CTR over damage. The reason is that damage UVs are monster-specific --- slimes, gremlins, etc. --- and never apply to Vanaduke or PvP. Hence, it is preferable to get "blanket" damage bonuses through armor and trinkets.
Second, if you're just talking about abilities, that aren't necessarily UVs, then it depends somewhat on the weapon and your playing style. Damage bonuses seem to have a larger effect (about 5% or even 7%) than ASI or CTR (about 3.5%), giving them an advantage. However, the practical effect --- number of hits needed to kill an enemy --- varies widely. Damage bonus may not lessen your hits-to-kill at all, or it may lessen them dramatically. It depends on the monster, depth, party size, status, previous damage, etc. In contrast, ASI and CTR always help you a little bit.
Now, about your specific question: Antigua-line guns have unsafe charge attacks, so charging is not common, so CTR is deprioritized. That's probably why you didn't ask about it. You can get data on Antigua ASI and damage effects from Lancer Knightz. Antigua guns fire so quickly, that having high ASI tires out your fingers pretty quickly. So here's my conclusion:
* If you're talking about family-specific damage bonus (from a UV, from Volcanic Salamander Suit, etc.), then go for ASI, not damage.
* If you're talking about blanket damage bonus (against all enemies), then go for damage, not ASI.
Edit: I also agree with Juances.
Antiguas have depressingly low damage per hit. That means damage bonuses are better for it. Doubly so if you're using it on neutrals.
I had some moderate success using a click-delay-release-click-delay-release etc of shooting with antigua lines, getting shots in during the click and a shot from the release. There's definitely some timing to it, but I found myself getting a higher rate of fire this way than spamming clicks. And less tiring.
Either way, without knockback or splash damage, the antigua lines need damage. Badly. And even then, they're terrible as main weapons, when the majority of late game battles are against mobs. Best used to finish off low HP monsters at a distance, where you can finish them off within a single clip.
Antiguas are best reserved for single targets or for chip damage/utility.
If you're a Gunslinger, you'll want to focus on it's single target strengths, which means you want damage.
If you're a Swordsman or Bomber, you'll want attack speed.
They benefit from both greatly, but if you have to pick one I'd go for damage.
Neither. It's a sidearm for wolvers and bombers. A gunslinger would be using alchemers. >:D
Anyways, it depends on the amount. Antigua already has a damage bonus agains gremlins/undeads. Stacking too much of that will make it go to waste a little (depending on the level). You'll want 2 damage meds, and the rest should be ASI for general use.
Remember you can have swifstrike, 2 pieces of armor, sprite perk and 2 trinket slots. It's easy to max both (not even counting UVs!).