I just ran into a scenario in the Vidpunktoj forum where I was uncertain of the exact difference between these two endings. I know -ado is normally used for -ing and -ation, to substantivise a verb, but I am unfamiliar with how -iĝo does the same.

The phrase being translated is:
"What is the burgling of a bank to the founding of a bank"

Another user translated "burgling" and "founding" as "enrompiĝo" and "fondiĝo", where I lean towards "enrompado" and "fondado". In one case we have the noun versions of "being-burgled" and "being-founded", which I would liken to "burgl-ation" (yuck) and "foundation". While in the other case "-ado" can represent both senses (-ation and -ing).

Is anyone with more knowledge than I capable of describing the real difference here? Are these two endings interchangeable?

While I hope that a spertulo will clarify the accuracy of -iĝo in this particular case, I believe that -ado is wrong here. Because the founding of a bank occurs only once at a certain time and is not a continuing action which -ado would imply.
That is why I didn't use -ado. But I am just a komencanto. =)

It is -in my humble opinion- a common misconception that -ado equals to English -ation and -ing.

Zamenhof defines ad' as follows:

Z:ad marque durée dans l’action; ex. pafo coup de fusil ― pafado fusillade | denotes duration of action; e.g. danco dance ― dancado dancing | bezeichnet die Dauer der Thätigkeit; z.B. danco der Tanz ― dancado das Tanzen | означаетъ продолжительность дѣйствія: напр. iri идти ― iradi ходить, хаживать | oznacza trwanie czynności; np. iri iść -- iradi chodzić.

As ustra has pointed out, -ad- conveys the meaning of duration. Mi dancis = I danced. Mi dancadis = I danced for a long time. danco = a dance. dancado = a dance that goes on for quite a while. Cxacxacxa estas danco. vs La hierauxa dancado lacigis min.

English gerunds (as in your example burgling and founding) can often be translated as a simple infinitive in Esperanto (the gerund originally is the inflected form of the infinitive and both are nominal).

Dancing is fun. = Danci estas amuze.

Concerning igx':

Z:iĝ se faire, devenir...; ex. pala pâle ― paliĝi pâlir; sidi être assis ― sidiĝi s’asseoir | to become; e. g. pala pale ― paliĝi turn pale; sidi sit ― sidiĝi become seated | zu etwas werden, sich zu etwas veranlassen; z. B. pala blass ― paliĝi erblassen; sidi sitzen ― sidiĝi sich setzen | дѣлаться чѣмъ нибудь, заставить себя; напр. pala блѣдный ― paliĝi блѣднѣть; sidi сидѣть ― sidiĝi сѣсть | stawać się czemś; np. pala blady ― paliĝi blednąć; sidi siedzieć ― sidiĝi usiąść.

Mi lacis. = I was tired.
Mi lacigxis. = My state changed to tired. = I got/grew tired.
Mi lacadis. = I was tired a long time.
Mi lacigxadis. = I grew tired continously.
lacado = the act of being tired a long time
lacigxo = the act of becoming tired
lacigxado = the continuous (prolonged) act of becoming tired

To answer your questions:
1) -ad- denotes duration and -igx- denotes a change of state.
2) No they are not. Their meanings aren't even remotely related.

Yes, -ado is not applicable to the sentence, as the burgling and founding are one-time events. While there may be instances where an English word with an -ing ending may be appropriately translated into Esperanto with use of -ado, it's not every time. You need to consider whether there is some significant passage of time in order to use it.

Enrompado would mean an ongoing burglary or a series of burglaries, that the rompŝtelistoj/burglars are perhaps taking out sums of money from the bank a number of times over a period of time.

Also looking at the original text of the Dreigroschenoper, the -ad- can't be justified:

Bert Brecht:Was ist ein Dietrich gegen eine Aktie?
Was ist ein Einbruch in eine Bank gegen die Gründung einer Bank?
Was ist die Ermordung eines Mannes gegen die Anstellung eines Mannes?

"Einbruch" is a burglary, a one time event; "Gründung" is a foundation/establishment, also a one time event. Neither of these words conveys any meaning of prolonged duration or repeated occurrence.

I personally would translate it like this:

Kiel malsamas serurrompilo kaj akcio?
Kiel malsamas enrompi en bankon kaj fondi bankon?
Kiel malsamas mortigi viron kaj dungi viron?

or

Kia malsameco estas inter serurrompilo kaj akcio?
Kia malsameco estas inter enrompi en bankon kaj fondi bankon?
Kia malsameco estas inter mortigi viron kaj dungi viron?

I seem to recall a thread in the English forums a while back that went into good detail on gerunds and the like, and there was a link to the PMEG where it said it is incorrect but very common to use infinitives after verbs (except for the accepted ones like voli, petei, ktp.), so "Mi sxatas kuri" should be "Mi sxatas kurado". I recall spending much time in the English and Esperanto wikis learning about gerunds and the mechanics that replace them in Esperanto.

That issue is separate from mine, and I have found the relevant section in PMEG to clarify the meaning. I really do need to read through that whole thing sometime.
You guys are correct, ado is only correct for a continuing action, and igxo for a one-time event. I suppose the part that hinders me is that English lumps everything under "-ing" quite nicely, but Esperanto is more specific. I find I have some issues with trying to apply English and Spanish grammar concepts to Esperanto.

That being said, the dictionary gives "-ing" and "-ation", and I would say that the "foundation" of something (or even the founding) is a one-time event. I suppose in English the verb dictates whether the action is an ongoing or one-time event. An example I've thought up would be "I like running" (continual/ongoing) versus "The running of the marathon was fun" (one-time).

PMEG:IĜ kutime aperas kun verba finaĵo, sed povas ankaŭ havi O-finaĵon, A-finaĵon aŭ E-finaĵon. Tiaj formoj montras la saman agan signifon kiel la verba formo: akriĝo = “la ago akriĝi”, akriĝa = “rilata al la ago akriĝi, akriĝanta”, sidiĝo = “la ago sidiĝi”, sidiĝe = “rilate al la ago sidiĝi, sidiĝante”, edziĝo = “la ago edziĝi”, edziĝa = “rilata al la ago edziĝi”, unuiĝo = “la ago unuiĝi”.

nornen:

Bert Brecht:Was ist ein Dietrich gegen eine Aktie?
Was ist ein Einbruch in eine Bank gegen die Gründung einer Bank?
Was ist die Ermordung eines Mannes gegen die Anstellung eines Mannes?

I personally would translate it like this:

Kiel malsamas serurrompilo kaj akcio?
Kiel malsamas enrompi en bankon kaj fondi bankon?
Kiel malsamas mortigi viron kaj dungi viron?

or

Kia malsameco estas inter serurrompilo kaj akcio?
Kia malsameco estas inter enrompi en bankon kaj fondi bankon?
Kia malsameco estas inter mortigi viron kaj dungi viron?

Your translations definitely convey the idea correctly but for my taste they differ too much from the words Brecht used. I am aware that translational work is not just done by translating word for word. However, these 3 sentences are simple enough to stick to original tightly. I am writing that because the structure you chose can be also easily replicated in German:

1) Wie unterscheiden sich x und y? (How differ x and y?)
2) Was ist der Unterschied zwischen x und y? (What is the difference between x and y?)

My point is that Brecht could have used these words but obviously didn't. That's why I'd prefer a stricter translation:

Kio estas serurrompilo kontraŭ akcio?
Kio estas enrompiĝo en banko kontraŭ la fondiĝo de banko?
Kio estas mortigo de viro kontraŭ la dungo de viro?

kaŝperanto:I seem to recall a thread in the English forums a while back that went into good detail on gerunds and the like, and there was a link to the PMEG where it said it is incorrect but very common to use infinitives after verbs (except for the accepted ones like voli, petei, ktp.), so "Mi sxatas kuri" should be "Mi sxatas kurado". I recall spending much time in the English and Esperanto wikis learning about gerunds and the mechanics that replace them in Esperanto.

I suppose you wanted to say: Mi sxatas kuradon. I do not second your opinion that "mi sxatas faradon" should be preferred over "mi sxatas fari". The tekstaro is full of examples (349 to be precise [1]) for sxatVF ww+iW. As infinitives are nominal in nature (as are participles and supina) they can stand (almost everywhere) where you would expect a noun. For example as a subject "Resti kun leono estas dangxere." or an object "Mi sxatas nagxi."

One example from the tekstaro is:

En la familia rondo li ŝatis legi laŭte el la versaĵoj de la poeto rusa Nekrasov, antaŭmajstro de Gorki.

In my opinion the PMEG is the best Esperanto grammar that I have seen so far, and I would be really shocked if they advocated against using infinitives as direct objects. Could you link to the article that states this?

----
I found only this. And this doesn't say that you shouldn't use an infinitive as a direct object after arbitrary transitive verbs.

ustra:My point is that Brecht could have used these words but obviously didn't. That's why I'd prefer a stricter translation:

Kio estas serurrompilo kontraŭ akcio?
Kio estas enrompiĝo en banko kontraŭ la fondiĝo de banko?
Kio estas mortigo de viro kontraŭ la dungo de viro?

I am a bit confused about your usage of -igx-.

Oni mortigas viron ---substantiviginte---> mortigo de viro (sen igx).
Oni dungas viron ---substantiviginte---> dungo de viro (sen igx).
Oni enrompas en banko ---substantiviginte---> enrompigxo en banko (kun igx).
Oni fondas bankon ---substantiviginte---> fondigxo de banko (kun igx).

Kial vi enmetis -igx- en la lastajn du, kvankam vi ne enmetis gxin en la unuajn du?

nornen:
Oni enrompas en banko ---substantiviginte---> enrompigxo en banko (kun igx).
Oni fondas bankon ---substantiviginte---> fondigxo de banko (kun igx).

Oni enrompiĝas en bankon , oni enrompas sin en bankon ---substantiating---> enrompiĝo en bankon
Oni fondas bankon ---substantiating---> fondado de banko.
Banko fondiĝas ---substantiating---> fondiĝo de banko.