I've always thought of them as historical references. I guess they are also monuments but, unless there's some shining praise inscribed, they seem like historical reference. And, of course, when Saddam put one of himself on every street corner, it's a monument.
To be honest, I don't know what statue caused all the problems in VA. Or which ones are being pulled down illegally or taken down under the cover of darkness by city/state.
But how in the hell does W.T. Sherman have a statue if they're monuments? That bastard was a terrible human that destroyed millions of dollars of property and killed women/children.