The problem can be seen in "OPTIMIZATION" section of main.jl where first callback_const_gen and then manual_const_gen are called.
callback_const_gen only adds 150 constraints, whereas manual_const_gen adds 451. Because of this, callback_const_gen doesn't achieve the true optimal solution, but rather one where not all necessary constraints have been added (all added constraints are satisfied).
Inside callback_const_gen, when split_constraint_callback is called the last time, it adds a constraint, as can be seen from the console output. However, the last call of split_constraint_callback should not find any more violated constraints. If a lazy constraint was added, then the solver should reoptimize the problem and then call the callback function again. Only after not adding any more constraints should the callback function not be called again.
Besides using MathOptInterface callback, these two methods should be equal and produce the same solution.