As young children, future Sparks show no sign of their future talents/pathologies. Usually in their late teens (with two known exceptions: Agatha at age 5 and Gil at age 8), they experience a Breakthrough, in which they suddenly create some invention, often violent. Thereafter they gain skill over time, like the rest of us, but with no dramatic changes.
That's for most Sparks. A few, though, experience a second breakthrough. This apparently doesn't happen spontaneously. In the two cases we know about, Albia was assisted by the other queens after taking possession of the Mirror in England, and Snacky needed the unwilling of Red, an extradimensional being brought to our three dimensions by a complex apparatus built by the Grey Hoods, his helpers in this work. (Albia can , so her ascension probably also involved interaction with higher dimensions.)
The first breakthrough happens entirely in the mind/brain. The new Spark shows newfound skill, and often newfound hostile intentions, but they are physically unchanged. The second breakthrough is different; most importantly, it makes the Spark immortal. The God-Spark also displays a halo made of symbols of some kind, different for each of them. Finally, the God-Spark has seemingly magic powers (levitation, teleportation, killing enemies with sheer energy, changing size) that Snacky deploys immediately after his second breakthrough, without having built any technology to enable them. (Klaus that "it is not magic." But we don't know his basis for that opinion.)
Constructs, even those hosting a sparky mind like "Lunevka", are unable to experience a second breakthrough. According to Lucrezia-in-Agatha, it takes an organic mind to "find the Spark, and build it into a flame - An Inferno." Whilst ascending, she exclaims "the neural pathways are already there, just waiting", possibly implying the groundwork to the second ascension is already within the brain of any Spark - although Agatha has previously "looked out at the infinite", which may be Albia's way of implying that an incident under Castle Heterodyne was involved.
However, this requirement for an organic mind (brain) turns out to be a lie as.
For a list of characters achieving a second breakthrough, see Ancient God-Queen.