Why can separating data into groups be confounding?

Example:

Simpson's Paradox

Think of a 2 x 2 table. Now separate into two groups (gender is a good one) and separate into two 2 x 2 tables, one for males and one for females.Odds of developing aortic stenosis among smokers relative to nonsmokers:

Males: OR = 1.23

Females: OR = 1.19

Males & Females combined: OR = 1.47