Why are the tensions the same for BC and BD?

They are at the same angle to the horizontal and experience the same force.

They are at the same angle to the vertical and experience the same force.

Actually why does the angle to the vertical affect the tension - I thought it only affected vertical and horizontal components of the tension.

This is what you kindly advised today with one of my other questions "the tension doesn't change but the VERTICAL COMPONENT OF THE TENSION must change if the angle changes."

In this case the vertical component must be constant, so the result force (the tension in BD or BC) changes to ensure the vertical component is constant. The vertical component in each string is equal to the tension in the light string.

In the other case, the resultant must of remained constant, so to ensure the result was constant when the angle changed, the horizontal and vertical changed correspondingly.

Isn't it the vertical component in each string combined that is equal to the tension in the light string though?

In this case the vertical component must be constant, so the result force (the tension in BD or BC) changes to ensure the vertical component is constant. The vertical component in each string is equal to the tension in the light string.

In the other case, the resultant must of remained constant, so to ensure the result was constant when the angle changed, the horizontal and vertical changed correspondingly.