The terms "comfort/stretch waistband" and "elastic waistband" are often used interchangeably. Both describe a waistband that will give. A comfort/stretch waistband can be elasticised at the sides, at the back, or all the way round. An elastic waistband more generally indicates that the waistband has been constructed to be flexible.
For smart men's trousers, an integrated comfort/stretch waistband is often ideal, as it gives freedom of movement without being obvious.