For the two-variable case, what are the practical differences between using the Engle-Granger procedure versus the Johansen test for cointegration? Is one universally more powerful than the other? Will one give more false positives or false negatives than the other? Should Johansen always be preferred?
I don't think that there is such clear answer as one of the test being uniformly more powerful/better sized than the other. As happens often in statistics, each test will perform better in a certain context, and bad in another; and since you don't know what is the exact DGP, these results are of little utility. In any case, Gonzalo and Lee (1998) end up recommending using both.
Rather than EG versus Johansen, the main question is whether you use the correct specification of deterministic components and lags. Using a badly specified test will probably be more harmful than using a "bad" test on the correct model.
Note also that the Philipis Ouliaris (1990) test is an improvement over the EG one, taking into account the supplementary variability from the fact that the residuals are created from estimated instead of true long-run parameters, as well as being invariant to normalisation of the cointegration relationship (i.e. which variable is taken as y). So in this case I would assume you are better off using the PO test in most cases.
As reference, I would recommend having a look at two early comparisons (although I am sure newer have come comparing fancier tests):
Jesus Gonzalo, Tae-Hwy Lee (1998) Pitfalls in testing for long run relationships, Journal of Econometrics 86 (1998) 129-154
Haug (1996) Tests for cointegration A Monte Carlo comparison, Journal of Econometrics 71 (1996) 89- 115
Hope this helps