Normally distributed and uncorrelated does not imply independent

Normally distributed means that a bunch of outcomes have an equal chance of happening. For example, if you roll a die, the chances of any number showing up is the same- 1 in 6. This is normally distributed.

Uncorrelated means that certain outcomes don't happen together. For example, if you roll a die, the chances that 3 and 4 both show up is the same as the chances that 4 and 5 both show up- it's very low. This is uncorrelated.

Independent means that one outcome does not depend on another outcome. For example, if you roll a die, the chances that 3 shows up will not make it more or less likely for 4 to show up. This is independent.

Normally distributed and uncorrelated does not mean that the outcomes are independent. Even if they have an equal chance of showing up and they don't usually happen together, one outcome could still affect the other - like if 3 and 4 often show up together.