Yes, it is different. The Root Mean Squared Error is exactly what it says.
(y - yhat)
(y - yhat).^2
mean((y - yhat).^2)
RMSE = sqrt(mean((y - yhat).^2));
What you have written is different, in that you have divided by dates, effectively normalizing the result. Also, there is no mean, only a sum. The difference is that a mean divides by the number of elements. It is an average.
Thus, you have written what could be described as a "normalized sum of the squared errors", but it is NOT an RMSE. Perhaps a Normalized SSE.