ARMA model vs ARIMA model Suppose I use the same p and q, are the two models below theoretically the same?

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*ARMA(p,q) on 1st order differenced y

*ARIMA(p,1,q)

I am aware of this post. But I would like to know are they conceptually the same model besides the minor differences in numerical optimization?
Also, if I run these two models, can I compare the two models say through RMSE based on the predicted values or through AIC/BIC of the model? Are they comparable as the y in the two models are not the same?

Note that we can define an ARIMA(p,1,q) model as follows:
$$\phi(L)\Delta y_t = c + \theta(L)\epsilon_t,$$
where the solutions of $\phi(z) = 0$ and of $\theta(z)=0$ lie outside the unit circle. As a consequence, $y_t$ admits a stationary and invertible $ARMA(p,q)$ representation after differencing once. To see this better, suppose that you have a series $y_t$ and you take the first difference. You would get a new series $x_t \equiv \Delta y_t$. At this point, considering an $ARIMA(p,1,q)$ model for $y_t$ or an $ARMA(p,q)$ for $x_t$ is exactly the same.
