When we should use $R^2$ instead of adjusted $R^2$?

Generally speaking you should use adjusted $R^2$ almost always if you are using it to decide whether to include additional regressors.

The reason for this is that unadjusted $R^2$ will never decrease when you add more regressors to the model (see Verbeek A Guide to Modern Econometrics pp 22). Adjusted $R^2$ solves for this issue as it penalizes for the additional regressor since adjusted $R^2$ is given by*:

$$text{adj } R^2 = 1 - frac{1/(N-K) sum e_i^2}{1/(N-1) sum (y_i-bar{y}_i)^2}$$

where, $N$ is the number of observations, $K$ is the number of independent regressors, $e$ are errors and $y$ is the dependent variable.

The inclusion of $K$ (unadjusted $R^2$ has there only 1) will 'punish' you for frivolously adding new regressors because now any time you add new regressor the $R^2$ will be lower. Consequently, this is a good safeguard against overfitting because if you add new regressor and you see $R^2$ falling that tells you the regressor you added does not improve fit sufficiently for it to be added.

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_{* Although note there are more than one adjusted $R^2$, but typically when people talk about adjusted $R^2$ they mean the one above that is used quite widely.}