The models are the effectively the same. The only difference lies in how you set up the contrasts. With the first model, the coefficients represent the average expression in each group. Thus, you'll have to use makeContrasts to set up comparisons between groups, and supply that as contrast in glmLRT. With the second model, the coefficients represent log-fold changes of particular groups over the group chosen as the intercept. As such, you can drop them directly with coef if you want to compare to the intercept group. (Of course, if you want to compare two non-intercept groups, then you'll still have to use makeContrasts.)
Provided you perform the same DE comparison, both parametrizations should give you identical results. I find the first model (i.e., the intercept-free approach) a bit easier to interpret in general, but that's my personal taste.