Normality is not product-closed
A product of normal spaces need not be normal.
Since the property of being a completely regular space is closed under products, this gives a proof that the property of normality is strictly stronger than the property of complete regularity. Specifically, any example of a product of normal spaces which is not normal, is also an example of a completely regular space which is not normal.
Stronger properties which are closed under products
Although normality is not closed under taking products, there are a number of properties slightly stronger than normality, which are closed under taking products. For instance:
Although a product of paracompact Hausdorff spaces need not be paracompact Hausdorff, it is true that a product of a paracompact Hausdorff space with a compact Hausdorff space is paracompact Hausdorff, and hence normal. Thus, paracompact Hausdorff spaces are binormal.