There's overlap, but the words are fundamentally different in meaning. "Quantified" implies a numerical approach, where you're simply trying to gauge one or more particular qualities.
"Assessed" implies a more holistic approach; you're trying to observe X as a whole, rather than zoom into particular aspects.
EDIT: Look at the example for "quantify" on Google:
It's very hard to quantify the cost.
This example makes sense, because cost is a quantity. You can measure it directly and refer to it in terms of numbers. On the other hand, look at Google's sentence for "assess":
The committee must assess the relative importance of the issues.
Here you see a weighing of various qualities and quantities. It's the bigger picture. Note that from the second example, you see that "assess" can have a similar meaning to "quantify" when referring to money:
The damage was assessed at $5 billion.