PIF is strictly worse than IGG. How many times have you seen a turn where you have more than 3 rituals in your graveyard that IGG isn't enough. If in any situation PIF is better than IGG that situation is win more rather than just win.

TES doesn't need a comeback. Any situation looking past LED is a lost cause. There is no reason to attempt going off in a situation using IGG, If IGG isn't a safe answer. If you are playing TES instead of ANT you should have them under a chant or if you are playing ANT you should have cast a few duress effects and know what you still have to battle through. If 5 storm isn't enough to finish an infernal tutor loop to kill them then you probably aren't winning anyway.

If you are low on cards and life TES was dead. No IGGY, AN or Diminishing Returns can safe you here but PiF is able to do so.

Remind that if you draw it past LED, you can pop it and flashback it instead. Even if you wish for it and it got countered you have another try.

IGGY gives you additional 5 storm if your combo out but doesn’t help you get to this point unlike PiF. It’s not THE no1 card in TES but a sure no2

As a guy who's tested Past in Flames in a combo deck, the card is not better than Ill-Gotten Gains, especially in decks that run Burning Wish. You either have to have an additional Burning Wish or Infernal Tutor in your hand/graveyard when you resolve it since Burning Wish exiles itself. Ill-Gotten Gains recurring Lion's Eye Diamond is one of the most powerful thing a combo deck can do in this format. This is especially true if you're playing a five color deck like TES since its usually protected by Chant so it doesn't care if it casts Ill-Gotten Gains. The inclusion of Past in Flames and Ill-Gotten Gains as wish targets seems redundant to me. If you're running chant effects, this even more true. You also needless create another weakness for your deck since two out of your three Storm engines are now vulnerable to graveyard hate. Your better off running Diminishing Returns; if you have at least two mana floating in some combination of blue, red, or black, you're running at better than 50% odds that post Diminishing Returns you'll win. @Gamble4Value is right, if you're worried about recurring countermagic, your better off casting Ad Nauseam or Diminishing Returns.

You'd probably run both in the sideboard but just Past In Flames in the deck, otherwise Ad Nauseam gets hurt somewhat. Wishboard looks like this:
1 Empty the Warrens
1 Tendrils
1 Past in Flames
1 Ill-Gotten Gains
1 Eye of Nowhere (wishable bounce)
2-3 other utility cards?
7-8 "normal" sideboard cards

That doesn't seem too bad.

Or are you implying playing another 4cc spell in the deck?

So, you're implying that it's better when you have Multiple Rituals? Fair, but you need 3 rituals in order to continue to go off. That makes 3, 5, or 7 mana from rituals (Dark) – Infernal Tutor (1B) – leaving you with 1, 3, or 5 for the tendrills. Additionally you need a mana floating and, if your Graveyard is light, you had better not be using Cabal's as you may lose Threshold and be producing less. This option leave you with Tendrills, the decks preferred win condition (as it wins there, not the next turn).

Right of flame only works if you have multiples in the yard, but you'll be dropping how much mana you can make each turn (Since they don't count their selves towards the bonus as they're on the stack – Doesn't work like Gravetroll and Dread Return). So that makes 2, 4, or 7 mana per copy in the yard respectively. Seems like that's more work for less payout unless all 4 are in the yard. Again, this is assuming you are floating the mana required to play the first. Also note, This way also required the Empty the Warrens or Grapeshot win (Which if you're looping this, you're probably already behind so This option pretty much is gone) unless you have access to BB.

No matter which way you cut it, LED is just better. With no mana floating you produce 3+X where X is equal to the other ramp spell's gain – Casting cost. (Dark Ritual is 3-1 totaling 2 additional mana). Double LED is 6 mana no matter what, and paired with Infernal Tutor is enough to Tutor for Tendrills and play it (usually resulting in a win). If you are using another ramp, you are required to have the additional mana floating, or take another route… This usually leads to your own death.

Hope that's helpful. Nice article Josh.

If you cast Past in Flames you can flashback more than 3 cards, which means you can get more rituals – which means you aren't necessarily losing mana vs what IGG would get you by returning LED.

You certainly aren't taking LED out of the deck – it plays far too well with Infernal Tutor and Burning Wish, and you can even use it to make mana for PIF if you have to.

More accurately, while PIF is close to IGG, Rite of Flames isn't even close to LED.

Generally how many times are you casting IGG without getting an LED anyway? Or even Duress, Cabal Ritual, Infernal Tutor? I just don't think Past In Flames is even close to IGG.

I disagree. I think IGG is better because it allows you to recur an LED. If your playing TES you probably have them under a chant anyway so it wont matter. In straight up ANT the IGG isn't generally the route to go against blue decks anyway unless you can get Duress, LED, Tutoror and beat their Force, Blue card, X.