Both Product Builder and Product Options Premium install and function in similar ways. You can set up a product customizer that allows the customer to choose different selections and options that will or will not add a price to the product with both. Both also allow for steps and options to be conditional on prior selections.
Note: As of February 19, 2021, Product Builder is no longer available to install on the Shopify App Store.
Builder does this by creating its custom product page for the product to be customized with a pretty structured, step-by-step customization process. The steps are groups of options that a customer proceeds through and makes selections, one at a time. Builder allows for images with each selection, and a separate preview image based on the combination of choices. Builder allows the price to be adjusted by selections.
Options uses a more fluid and unstructured product customization process. Options are created and paired with products in the form of relationships. You can use the same options with multiple products and you have more control over how selections for options are made, whether it be in the form of a drop-down menu or radio buttons. You cannot associate pictures with options and they are set up directly on Shopify's product pages. Options adds its priced options as an add-on to the main product price, and while the add-on price is visible on the main page, it will not change the main product price on the product page without a custom modification.