Html.TextBox is not strongly typed and it doesn't require a strongly typed view meaning that you can hardcode whatever name you want as first argument and provide it a value:
<%= Html.TextBox("foo", "some value") %>
You can set some value in the ViewData dictionary inside the controller action and the helper will use this value when rendering the textbox (ViewData["foo"] = "bar").
Html.TextBoxFor is requires a strongly typed view and uses the view model:
<%= Html.TextBoxFor(x => x.Foo) %>
The helper will use the lambda expression to infer the name and the value of the view model passed to the view.
And because it is a good practice to use strongly typed views and view models you should always use the Html.TextBoxFor helper.