In Vipassana, what's the difference between 1st and 2nd "knowledge of dissolution" (paṭhama- & dutiya- bhanga-ñāṇa)

Bhanga-ñāṇa or "knowledge of dissolution" is the first major direct (nonconceptual) experience of Emptiness. Although every practitioner's experience is unique, there are some common themes that can help one recognize the milestones as they are passed. Here is one description of bhanga-ñāṇa by Patrick Kearney of buddhanet.net, a dharma teacher in the lineage of Mahasi Sayadaw:

What happens next? The meditator’s awareness and concentration continues to develop. As a result, he now sees only the passing away of phenomena. It is as if his awareness is so fast, it is faster than the experiences he is examining, As soon as he places his attention on some aspect of his experience, it disappears. This is the knowledge of dissolution (bhanga-nana). In a weak aspect, this can take the form of the meditator apparently losing his concentration. It seems like he can no longer focus on anything; his attention keeps sliding off whatever he tries to look at. It can be lie trying to grasp something that slips out of your hand the moment you touch it. In a stronger aspect, it can be like falling into the black hole of Calcutta. Wherever you look, there is nothing - only blackness. The meditator is shocked, because he used to be able to focus on anything. Now, it seems, he can focus on nothing at all. All his good work has dissolved into nothing.

In reality, this and the following several stages are a very painful experience. It feels like losing your mind, so frustrating it is. You can't even focus on your frustration -- the moment you look at frustration, it slips right through your fingers. This is truly the very bottom of "the dark night of the soul". All preconceptions collapse. You really see the BS-ness, the emptiness of all mental formations. Up until this moment you were holding onto the hope that something must be real, but now your ego is exhausted and can no longer maintain yourself.

As for the distinction between the first and the second "knowledge of dissolution", when the thought-form you are "looking" at collapses, falls apart, and dissolves right in front of your "eyes" -- that's the first bhanga-ñāṇa, and when, in utter frustration and disbelief, you look at this "looking", and it itself collapses, falls apart, and dissolves -- that's the second bhanga-ñāṇa.

In case you don't believe my first-hand experience ;) here is an explanation by Ven. Sayadaw U Jotika in his "A Map of The Journey" (a transcript of a series of preparatory talks given prior to a meditation retreat held in Australia):

At this stage, every time the meditator is meditating -- no matter what the object is, even the movement, (without paying attention to the shape anymore but to the sensation) - the meditator is aware of the sensation and the very fast passing away one after another very clearly. All the shapes and solidity disappears, which means that you don't pay attention to shape or solidity anymore. You pay attention only to sensations and their passing away very quickly.

Seeing the object passing away very quickly is the first bhanga-ñāṇa, paṭhama-bhanga-ñāṇa. Seeing the wisdom passing away is the second bhanga-ñāṇa, dutiya-bhanga-ñāṇa. The two together make the bhanga-ñāṇa complete.

In any insight there is the beginning and the maturity of insight. In the beginning you see the object passing away very quickly. When this becomes more mature, stronger, you see the passing away of consciousness, vipassana consciousness, and the wisdom also passing away together.

You notice one thing, that passes away and
that consciousness which is aware of
that passing away also passes away.

...

Because of seeing this passing away, passing away, continuously you feel that this process is dangerous; it is passing away so quickly you cannot rely on it. You cannot identify with it. You see that nobody can identify with this and hold on to it and rely on it. There is no reliance. There is nothing reliable in these mental and physical phenomena.

As interesting as it is to read about these things, my advice to all practitioners is to forget about these stages/levels and instead focus on your practice. In Buddhism, knowing stages does not help you get there. Buddhist experience cannot be contrived! Once every 5 years you might want to check yet another stages-of-the-path text, congratulate yourself on another achievement, confirm that your intuition was right again -- and then forget about stages for the next 5 years.