In discrete distribution when we plot PMF the Y axis is probability. In continuous distribution when we plot PDF the Y axis is density (probability is the area under the curve). So, we learn that density values are not probability values.
But what happens when I approximate binomial with normal distribution. Consider an example case of B(100, 0.5). So, Mu=50, sigma=5. I calculated both binomial and normal distributions with these parameters. Below is the plot. For binomial distribution my Y axis is probability but for normal distribution it is density. But numerically the values practically overlap. Obviously, probability in a point for normal distribution is still 0. Does this makes sense to you?