Html.Partial returns a string, Html.RenderPartial calls Write internally, and returns void.
The usage (using Razor syntax):
@Html.Partial("ViewName")
@{ Html.RenderPartial("ViewName"); }
The usage (using WebForms syntax):
<%: Html.Partial("ViewName") %>
<% Html.RenderPartial("ViewName"); %>
Will do exactly the same.
You can store the output of Html.Partial in a variable, or return it from a function. You cannot do this with Html.RenderPartial. The result will be written to the Response stream during the execution.
The same is true for Html.Action and Html.RenderAction.
