Say you have a case where you want to perform survival analysis on a set of individuals with a single binary-valued covariate. Assume that the PH assumption is satisfied. Say that the goal is to analyse the difference between the two groups (corresponding to the 2 values of the covariate). In this case, what would be the difference between using Kaplan-Meier and Cox PH model? Also, intuitively, why would they be different? I know that Cox takes into account the value of the covariate, but in this case, since it is binary-valued, Kaplan-Meier would also do this in a sense.
Well, you get a hazard ratio exp(beta) from the Cox model, showing the % increase/decrease in the hazard, while a K-M will give you an estimate of the survival curve only, i'm sure that there would be no difference in, say, doing a log-rank test on a K-M survival curve for your two groups (x=1 or x=0), and doing the wald test (or z test or whatever parameter test you like) for the beta(x) in the cox model, if one shows evidence for differences between the groups the other most certainly will too. To me it only matters if you want to interpret your effects on the scale of the survival curve (K-M) or hazard function (Cox).