A landing page is a page that has only one focus. It can be to capture an email address or to sell a product or something else. It doesn't have any menus or other buttons or links except the one that I just said so the visitor is not distracted elsewhere.
On the other hand, a website is a number of pages that someone can visit to do multiple of things.
So why don't you do both? You can promote the landing page through social media and also have a website to inform people of what you really do.