LoadUI and LoadUIWeb tackle different, if related, technologies.
LoadUI focuses on API load testing.
LoadUIWeb focuses on browser-driven, HTTP-traffic load testing.
If a website is designed such that it mostly makes calls to a certain API (most often REST APIs) and displays the results on the user's screen, then you have a good case to use LoadUI to test the API directly (because that's what you're interested in testing, not the ability of the browser to update records on the screen). In any other case, you should use LoadUIWeb.
If you are able to simulate API calls using one of the LoadUI Runner Components (Web Page Runner, HTML Runner, Geb Runner etc.), it may be a good choice for testing your API-driven website.
If the only way you can generate these API calls is by actually using a browser as a client and doing things manually, then clearly you should use LoadUIWeb.
If you don't know anything about APIs or your website is not actually driven by APIs, you should use LoadUIWeb.
Hope this helps.
Renato