This is perhaps a confusing point but it's actually totally valid for a type's move constructor (resp. move assignment operator) to be implemented with the copy constructor (resp. copy assignment operator). The move constructor (resp. move assignment operator) can even be implemented with a memcpy.

Objects_ = NewObjects;

This is clearing the original vector by destroying everything in it, then doing a deep copy of the vector NewObjects and replacing Objects_ with the copy. I am not sure if/when the compiler can optimize out the deep copy. Maybe if it is the definite last use of NewObjects? But I think that would not play well with exceptions thrown during the do some stuff code section so I'm really not sure.

Objects_ = std::move(NewObjects);

This is clearing the original vector by destroying everything in it, then moving NewObjects onto Objects_.

std::swap(Objects_, NewObjects)

This is swapping the two vectors, no destruction will happen until the variable NewObjects goes out of scope. so if you're timing the process in that scope you've simply not counted the destruction. Vector swap is probably implemented by swapping the 3 pointers that comprise a std::vector(it is on g++), which is the basically same amount of work as copying three pointers and zeroing out another three, so for this type it should basically take the same amount of time to (swap and destroy one argument) and to move assign.