What is the reason why ARIMA(0,1,0) on $y_t$ and ARIMA(0,0,0) on diff($y_t$) are not identical time-series models?
I don't know R code but are you estimating an intercept in the ARIMA(0,1,0) model? Because if not, this could be why there is a difference, since you are estimating an intercept in the ARIMA(0,0,0) model.