I would definitely NOT include boar spear or harpoon. Read the description of those weapons. Boar spears have a spiraled weapon head and a crossguard, and harpoons have a barbed head. Both of those weapons, by their very nature, are far too technically complex to have been available to a Neanderthal.
Great Club (and the regular ol' Club, of course) make perfect sense. A stone-headed hand axe, perhaps. Same thing with a stone-bladed dagger. Spears, certainly, either just as a sharpened wooden shaft or a stone blade.
While they couldn't make them, they could probably use most of the simple weapons if they took them off previous kills. After all, at it's essence, all a mace or morning star is, is an improved club. The skill to use them is the same.