@Html.Sitecore().ViewRendering("Path to the View")
will trigger mvc.renderRendering pipeline. And your view will be rendered in almost a same way if you add it to placeholder. Difference from Html.Partial is in cycle of processing your view. Rendered result could be different if you depend on something in that pipeline (e.g. Caching as mentioned @Gatogordo). (or if you added some processor there by yourself).
If you want you rendering be the same if you add them via placeholder then use Html.Sitecore().ViewRendering
For
@Html.Sitecore().Controller("Controller Name", "Controller Action")
and
@Html.Action("Controller Name", "Controller Action")
difference also is in it's execution lifecycle. Sitecore ones is executed via ControllerRunner that gets Controller from SitecoreControllerFactory and execute some action. ASP.Net MVC action is executed via HttpContext.Server.Execute and do actually the same. But looking on implementation I can make an assumption one of differences is in routing. In case of using ASP.Net MVC helper your route values can bring you to some Sitecore item rather that required controller action if it will match. Sitecore helper will always execute controller.
If you need more details you can open System.Web.Mvc.Html.ChildActionExtensions.Action and Sitecore.Mvc.Helpers.SitecoreHelper.Controller in reflector and compare them step by step.