When the stellar material hits the hot surface of the white dwarf, the material can fuse quickly and create what is called a nova. Novae are not as energetic as supernovae, but will increase the luminosity greatly. However, the material accreted is rapidly used up and the nova dies down. Nova can occur many times and do have a predictable period that can be measured.
However, if the material on the surface of the white dwarf accretes too fast, the material can increase the mass of the white dwarf to over the Chandrasekhar limit. When this happens, instead of a nova explosion, the white dwarf undergoes a catastrophic collapse. What occurs is a Type I Supernova.
Since the white dwarf mass becomes larger than 1.4 solar masses, the electron degeneracy of the white dwarf cannot overcome the gravity the white dwarf experiences. The white dwarf catastrophically collapses, allowing all the material in the white dwarf to fuse rapidly. The outward explosion from this sudden release of energy completely destroys the white dwarf and companion star. The way astronomers differentiate between Type I Supernovae and Type II Supernovae is the lack of hydrogen lines in Type I. Because the white dwarf has no remaining hydrogen and the hydrogen from the companion star is completely used up in the fusion process, we know that Type I Supernovae can only be created by the sudden collapse of a white dwarf that accretes material from a companion star.
From Wikipedia, OOCalc chart
Looking at the above image, another difference between Type I and Type II supernova is that Type I are typically much brighter in the beginning but fade much quicker.
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