-xtreg, re- allows to have an unobserved component in the error term which is random, then the loglikelihood has a composed error term that requires assumptions on the distribution of u(i) and e(i,t)... convention says that both are normal. Similar approach is applied in frontier models, but there the distribution of u(i) is half-normal or gamma.
-xtgls- is GLS for panel data, asumming some structure for the distribution of e(i,t). For example, in terms of variances, correlations (cross-sectional) or autocorrelations (time series). The default is no correlation accross panels, homoscedastic errors and no autocorrelation, ie iid errors across panels and time... the latter is the same assumption of the standard LS (-reg-). You can get the same standard errors than -reg- adding the option -nmk- to xtlgs as follows: