So what exactly is so good about it? A lot of people in-game and on the forums say its probably the best set in the game (or one of them), but every time I look at the stats, I just don't see it. So, what am I missing?

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1. The shields are quite good on their own.
2. Tons and tons of Subsystem Shutdown Protection, i.e. "Hot Restart" mechanics. It has this to quickly bring shutdown subsystems back online for Shields, Weapons after only 1 second. Nukara Space Set only has it for weapons and even then, it waits 3 seconds to do so.
3. More Bonus Defense. One of the few sets in the game that grant +Bonus Defense. High Bonus Defense means things miss you more. Things that don't hit you are repairs that you don't have to deal with.
4. Great performing Hyper-Impulse Engines are part of the set. With the high subsystem power levels across the board possible today, Hyper-Impulse Engines bring completely superior speed and turn rates. I have been turned away from sets because they slap a mere Impulse Engine or worse, Combat Impulse Engine as part of the set, bringing my ship handling performance down. For me, this is not an Escort/Raider issue. This is an issue for ALL my ships because I can achieve high Subsystem Power levels no matter what. Having high power levels, leaving Hyper-Impulse Engines for the sake of Impulse or bottom rung Combat Impulse Engines is a night and day affair.
5. Both Warp/Singularity Cores gain [AMP].
6. Outstanding 3 & 4 Piece Bonuses to increase team damage capability.
The Iconian Rep Space Set has lots of things geared for general combat purposes. It's not meant for Science Vessels, Science-heavy builds.
When the Iconian Reputation first came out, I openly derided the usefulness of the set. But when I began to really check the stats and bonuses, I did a complete 180. Several of my many characters between the Feds, KDF, Roms use the full set now. Breaking integrity of the set is unimaginable for me, even after the advent of the latest Reputation. The set has many things going for it and is worthy of consideration of using the entire space set, not just 1 or 2 pieces at most like most other sets do.
Other sets do some similar things, but for tactical high DPS builds it works quite well, that's why so many people like it; it makes things easier.
I use all or part of the set for a few builds/designs. An alpha strike of GW/TR+DRB+SSV+HY attacks generates a lot of attention, and the set has helped me deal with all that additional attention, and live. I've also used the set when trying to build a FBP sci ship; the four piece set was front and center when I ran a Scramble Sensors design for the Mirror Invasion event-the set often helped me survive the phaser barrage. I'll also use the Iconian two piece when I run the Reman two-piece running three plasma torpedoes fore supported by SSV, DRB, GW, or TR.
I was completely guilty of ignoring the set when it first came out, I'm a torpedo boat captain and never thought I'd use it. I was wrong.
Elements of the set also fit the way I pilot, as I nearly never stop flying to constantly discharge fore and aft torpedoes.
Star Trek Online volunteer Community Moderator
To clarify, the space set does not specify AP damage boosting. It's generic "damage" increases for Set 4 or Energy Weapons specific for Set 3 bonus.
Nailed it!
Nicely done sir.
Either way I still think some people put too much stock into it. great set for sure, just over hyped imo
Star Trek Online volunteer Community Moderator
I still vividly remember my many posts of me making fun of the Iconian Space Set when it came out. I am still eating my words about how wrong I was on that awesome set.
Some ideas: http://www.arcgames.com/en/forums/startrekonline#/discussion/1208919/buff-combat-impulse-engines
It's funny because a long time ago, Combat Impulse Engines were the most favored engines in general use. This was back when regularly high Subsystem Power Levels was very hard to accomplish or near impossible. Not unless the build was specifically shunted for high Engine Power would you ever slot Hyper-Impulse. Without all the power boosts possible today, most builds put Engine Power as the lowest priority. This is true now as it was long ago. But with the power boosts today, you can have high levels across all subsystems. All. The. Time. If you're Engineer, it's stupidly ridiculous. If you have OSS to spike it even higher past the caps, LOL!
I learned the value of Hyper-Impulse Engines because of PVP and transplanted that habit to PVE. The ease of freedom of flight is just as important in any form of combat as the build itself, at least to me.
Anyways, when Delta Rising came out, I was cycling my Fed TAC through the Fed ships for Starship Masteries. With a very quick, very hasty build on a Guardian, a sluggish ship to begin with, I was able to maintain regularly high power levels. I slapped on my very ancient Omega Hyper-Impulse Engines Mk XI and I could almost make the Guardian spin like a top in place with BFAW going like a Disco Ball. Well, okay, that's an exaggeration but with high power levels and Hyper-Impulse Engines, the slow Guardian behaves much faster, much smoother, reacts more responsively.
Slap a pathetic Combat Impulse Engine on that same setup, then it handles like a pig again.
Oi! What's wrong with that?
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Maybe there was a different time in the 2 years I wasn't playing. I wasn't around for the introduction from Legacy of Romulus - I still remember that it was impossible to put a power setting below 25. I believe LOR changed it to 15. Maybe then efficiency alone didn't suffice.
Some people want their engine to move them around and make hard turns in combat, instead of them spending all their time posting on the forums.
Don't let them promote you. Don't let them transfer you. Don't let them do anything that takes you off the bridge of that ship, because while you're there... you can make a difference.
-Captain James T. Kirk
Also, once upgraded to amp the full set will outperform a fleet core but it isn't cheaper by any means.