The recent changes to Superweapons, along with recent discussion about which is best, prompted me to do some research on them.  I tested each, noting the effects, recorded descriptions of each, and looked up old posts regarding them.  The following summarizes each weapons characteristics. 

First, the 1.2 changelog states:

• Novalith Cannon:
• Cannon shots now cause a planet to suffer 100% reduced trade income. This effect will no longer stack.

• Kostura Cannon:
  • Will no longer damage and stun enemy ships.

TEC: Novalith Cannon
  Primary Effect: 3500 planet health damage, and 150 population killed.
  Secondary Effect: Cannot re-colonize for 2 minutes "due to strong bombardment raditation".
  Tertiary Effect: Trade Income is reduced by 50% for 30 minutes. "Trade Income 100%" (in red).  
  Tertiary Effect: Population cap reduced by 40% for 30 minutes. "Max Population 40%" (in red).
  Target:  Only colonized enemy planets.
  Cooldown:  360 (6 minutes)  Antimatter: 200 per use.
  Cost:  8000 credit 600 metal 500 crystal per gun, 18 tactical slots.
     TL:  6000 credit 450 metal 375 crystal per gun, 15 tactical slots. (-25% cost, -3 slots).
  Research prerequisites:  3 Tier 2, 2 Tier 4, 1 Tier 6, 1 Tier 7, 1 Tier 8,  Military tree.
     TEC Loyalists prereqs:  only 2 Tier 6!, Military tree.
  Research cost: 8800 credit 1400 metal 2275 crystal
     TL Res. cost: 2800 credit   500 metal  800 crystal 



Advent: Deliverance Engine
  Primary Effect:  attacking ships gain 25% increase in damage for 150 sec., "damage output 25%" (in green)
  Secondary Effect: Powerful spread of '50 culture' from the planet causing a temporary loss of around 10% allegiance
  Tertiary Effect: Shield mitigation increased 6% due to required culture research, and culture's presence.
     Adv. Loyals: also recieve 20% phase missile block.  Add'l techs can also boost AL damage in culture.
  Target:  Colonized freindly & enemy planets.  Not neutral uncolonized gravity wells, nor dead roids (colonized or not).
  Cooldown:  360 (6 minutes)  Antimatter: 200 per use.
  Cost:  8000 credit 600 metal 500 crystal per gun, 18 tactical slots.
  Research prerequisites:  3 Tier 2, 2 Tier 4, 2 Tier 6, 1 Tier 8,  Civilian tree.
  Research cost:  8700 credit 1975 metal 2225 cystal



 Vasari: Kostura Cannon
  Primary Effect: 1250 damage to all structures (before armor and shield mitigation reductions)
  Secondary Effect:  Disable orbital structures for 3 minutes
  Tertiary Effect: The planet acts as if there is a phase gate at the location for 1minute 30seconds.
  Target:  Only colonized enemy planets.
  Cooldown:  360 (6 minutes)  Antimatter: 200 per use.
  Cost:  8000 credit 600 metal 500 crystal per gun, 18 tactical slots.
  Research prerequisites:  None!  1 Tier 8,  Military tree.
  Research cost:  1800 credit 350 metal 550 crystal 

Some additional observations:

The Novalith kills most planets with one shot. 2 shots to kill a HomeWorld (3 for fully upgraded HW in Rebellion). Multiple Novaliths can be a game winner if you methodically kill all.  Starbase upgrades (ie: Aux Gov) can prevent planet loss. 

Repeated staggered shots from 2 Deliverance guns on one enemy planeet, say every 3 minutes, will eventually bring allegiance to zero, and affect the adjoining planets as well.  Providing they don't have offsetting culture!  Shots stack, so its possible to have 100 culture bombarbing a planet (requires 5 unupgraded culture towers there to offset).  But that 100 is divided by the number of outgoing phase lanes reducing effectiveness against adjoining planets. 
(At the maxi. -0.07%/sec allegiance loss rate = about 6 minutes for 25% allegiance, or 24 for 100%);
(At the typical -0.04%/sec allegiance loss rate = 10.42 minutes for 25%, or 41.67 for 100%).

Three Kosturas shots in rapid succession will destroy extractors. It takes at least 5 shots to destroy frigate factories and turrets. About 6 shots to kill labs, trade ports and cap ship factories. Most structures have 5-7 armor. 5 armor reduces 1250 damage to 1000 effective damage. Most stuctures will exhibit around 900 damage. Since turrets have around 12 armor, they will exhibit damage in the high 700s for each Kostura shot. A single shot does little effective damage, unless it is followed by a fleet jumping through the temporary phase gate, and used to disable structures.

The Kostura only costs about one-quarter of the other 2 superweapons to research. And since it requires no other prerequite research, it can be completed much quicker. The Novalith and Deliverence Engine research are roughly comparable to each other in cost, and have fairly similar prerequisite research requirements. But the Deliverence Engine costs about 500 more metal, and it can be argued that the Deliverence Engine's cultural prerequisites are less useful.  TEC Loyalist Novalith research costs only about a third, and the gun itself costs 75%, while taking only 15 slots, (so it is possible to put 2 guns at most planets).

Each gun costs the same to build individually (except TL), and they all have the same cool down and antimatter requirements.

[Edit] Rebellion has introduced new features affecting superweapons.  For example:  TEC Loyalists can now buy Novalith's much sooner, and for less cost.  Also Advent Loyalist receive several new cultural boosts in the prerequisites leading up to a Deliverance engine. 

I have updated this post to reflect those changes.