There is absolutely a difference between middle and working class people. To suggest there isn't is a complete lack of understanding with regards to what it means to be working class.

By definition middle class people work, it's in the middle between people who have to work to survive and the independently wealthy. However, the middle class has the capability to build wealth, own a home, retire someday, leave a little over for their kids to get their lives started. Working class people are often life long renters, would go bankrupt if they got cancer or any other serious medical illness, would be in significant financial strain if their car broke down, they very likely do not have the means to save up 6 months worth of expenses as a safety net.

Yes, these terms are rather antiquated, but with each and every passing year, there is an increasing distinction between the professional & managerial class (often referred to as the middle and upper middle class), and the precariat class (the working class). Also note that the former makes up ~34% of the population, and the later makes up 45% of the population, with the 1% on the upper end and the truly poor on the other end.