I trained a linear regression model using 3000 data points. (OLS regression, no regularization.) Then I trained another model with the same predictors (about 25), but with a subset ($n=700$) of the 3000. Is there a systematic reason why the model trained with the subset of the training data performs significantly better on a test set ($n=1300$)?
You didn't really give enough information for a good answer. For instance, you didn't tell us how that subsample of 700 observations were chosen. So for this answer I will assume it was taken randomly. If that is incorrect, you should tell us.
So then, the reason was probably only luck. To see if that is true, repeat the subsampling randomly 100 times (say), and plot a histogram with the errors on the validation set. You should be able to conclude.