Which is better?

Both a presidio and Bastion give you influence points which will increase your BH's rank among the other BHs.

Presidios

At level 5 they offer everyone's troops a 25% defensive boost. This helps in both defending against King Carlos and mainly against those attacking you whether they be pot shots for pvp points or a rival BH trying to take your presidio. However a presidio can also gain more defense a max of 20% or an additional 10%. 

 20% - Gained by having a unified Presidio. To have a unified Presidio you must have control over 5 presidios. Basically 4 presidios surrounding one in the center. The center one will gain the defensive boost. This will also give you a 20% territory boost.

 10% - Gained by having 3 presidios in a vertical line. This bonus applies to the center presidio. Which also gives you a 10% territory boost.

Pardons - Kind Carlos will attack your presidio generally twice per day. Giving your BH pardons. Pardons give a stat boost to your troops and also cut rum consumption by 50%. Its best to start by pardoning Armada troops first. Remember to let your crew know that the reports they receive are going to everyone. And those pardons are shared among the entire crew.  Pardon cost can be reduced by 50% if you apply 50 pearls in the Witch Doctors Hut. A second way is by a BH achievement

BH Achievements - Presidios can help you gain BH achievements which give you influence points and depending on the achievement a defensive boost a percentage. Whether its for reaching a mile stone for taking pardons, capturing an enemy level 5 BH, having X level 5 presidios or for the amount of territory you control.

Bastions

Unlike Presidios the defensive boosts  for troops don't exist for bastions. However like a Presidio they also offer influence points to increase your BH rank. Upgrades though unlike presidios aren't by using resources. You must use Stones or Codexes

Influence Points - gained 2 ways. 1 is by the level of your Bastion. Each level increases the influence you gain from having a Bastion. Second way is your knowledge. It gives a percentage of influence boost but only applies when you have a presidio. 

Knowledge - You can upgrade your knowledge straight to level 20 on one stat if you choose but the knowledge level that applies is in direct correlation to the Bastion level. Each level in Offense or Defense increases the stat that it applies to - Armada, Fleet, Mercenary or Pirate. The last 2 knowledges are for Unit speed and Rum Consumption. The latter can be very important to your members that have massive armies and need help with controlling their rum consumption.  

Bastion upgrades - Until level 8 all upgrades require nothing but Stones, however once you reach level 8 if you'd like to upgrade farther it now requires you to downgrade a rival Bastion of level 7 or higher, plus the required Stones.

Bastion and Presidio

How does one help or hurt the other?

Bastion 

No benefit to the bastion from having a presidio.

Presidio

The bastion actually gives quite a bit of help. 

Knowledge - Your units are now stronger and receive a bonus in defensive stats no higher than your bastion level.Also your knowledge is also giving you a boost to your territory which is increasing your influence points. So if your land locked you now can still have a bit of territory that you may not of had without this boost. 

Which is better to have a bastion or a presidio or both? Can ye handle having both and still defend whats yours?