Not an expert on simplex, but here's my attempt on an answer:
In general, the solution of the (previous) LP Relaxation will no longer be primal feasible when the primal LP is tightened (e.g. new cut in branch & bound).
In the dual simplex, new primal cuts correspond to new dual variables, which are initialized as nonbasic, and thus the previous solution is still dual feasible. Thus dual simplex does not need to regain feasibility.
So in a mixed-integer-programming context, dual simplex will usually outperform primal simplex.
In general it is easier to get dual feasibility than primal feasibility, and dual simplex appears to make more progress in many iterations.
Many commercial solvers also offer Barrier methods to solve LP(-Relaxations). The big drawback of interior point methods is that they can't really be warmstarted, and when solving mixed-integer-programmes using branch & bound are generally given a reasonable warm/hot-start (the solution of the previous LP Relaxation).
I've seen a recommendation before that stated that one can use IPMs to solve the root node of the MIP, and then use dual simplex afterwards. See Jakob's answer on how to do that in Gurobi.