Are average velocity and average speed always equal?

Average speed and average velocity are not the same quantity, and they are calculated differently. This is not just because velocity is a vector and speed is not; it has to do with distance and displacement.

Average Speed

Average speed is not a vector quantity. It has no direction.
The simplest way to find average speed is to divide the total distance travelled by the total time taken.

Average Velocity

Average velocity is based on where she ended her trip; specifically, how far she is from her starting point. This distance from the starting point is called displacement, and it's a vector quantity.

Example

Lets say Shyam moves 5 m to his left and then moves 2 m to the right. Also, say he takes 2 seconds to do this.

Distance travelled by Shyam = 5+2 m = 7m

Time taken = 2 s

Hence, average speed = total distance/time = 7/2 = 3.5 m/s

But displacement of Shyam is not 7m. Displacement is the difference in the initial and final positions, and it also has direction. Displacement of Shyam is 3 m to the left, because in the end, he is 5-2 m = 3 m to the left of the position from where he started. And it is important to mention the direction when speaking about displacement. So it is important that displacement is 3 m to the left

Time taken = 2s

Average velocity = displacement/time = 3m left / 2 = 1.5 m/s to the left. Since velocity is also a vector, it is important to mention direction here also.