The method you describe is called random restart hill climbing (or sometimes shotgun hill climbing), and it is a different algorithm from simulated annealing.
Yes, generally as the number of iterations $k$ increases both methods will eventually give a location $w_i$ which reaches a global optimum $w^*$. This is for the simple reason that both incorporate random search. That is, a random restart (hill climbing) or random move (simulated annealing) can turn out to coincide with a global optimum. Nevertheless, here are two important differences:
- random restart hill climbing always moves to a random location $w_i$ after some fixed number of iterations $k$. In simulated annealing, moving to random location depends on the temperature $T$.
- random restart hill climbing will move to the best location in the neighbourhood in the climbing phase. In simulated annealing, the location is selected randomly, you always move if it's better than your current location but with some probability related to $T$ you may move even if it's worse.
Simulated annealing is a somewhat more complicated algorithm, and depends on the temperature schedule which determines $T$ at iteration $k$. If the temperature $T$ is set to a very small constant value then the simulated annealing becomes like stochastic hill climbing. If $T$ is set to a very large constant value, then simulated annealing becomes like random search. The way you select the temperature schedule determines how you navigate between these two different type of behaviour.
tldr: these are different algorithms, but they use similar ideas to incorporate random sampling into search.