Returns to Scale -> changes in output when factors of production are increased
Economies of Scale -> reduction in average unit costs as output increases.
-> I would say they go hand in hand - when there are increasing returns to scale, the %Change in output > %Change in input. This implies that the firm experiences economies of scale because, as it increases the scale of production, it becomes more efficient and can spread fixed costs over a larger output. In other words, as the input increases, output increases at a faster rate, leading to a reduction in average cost per unit of output.
This also works with constant and diminishing returns to scale. If you look at the standard diagram for economies of scale ( U shaped with the straighter line in the middle) you can identify:
-> increasing returns to scale & economies of scale as the first third (downward sloping LRAC)
-> constant returns to scale as the second third (straight line of the LRAC)
-> decreasing returns to scale & diseconomies of scale as the last third (upward sloping LRAC)