TLDR; While the formulas may be similar, RMSE and standard deviation have different usage.
You are right that both standard deviation and RMSE are similar because they are square roots of squared differences between some values. Nonetheless, they are not the same. Standard deviation is used to measure the spread of data around the mean, while RMSE is used to measure distance between some values and prediction for those values. RMSE is generally used to measure the error of prediction, i.e. how much the predictions you made differ from the predicted data. If you use mean as your prediction for all the cases, then RMSE and SD will be exactly the same.
As a sidenote, you may notice that mean is a value that minimizes the squared distance to all the values in the sample. This is the reason why we use standard deviation along with it -- they are related species!