Add-ons are external apps that provide external functionalities. those apps don’t need HA to function since they are external to HA.
integrations are apps(-ish) that operate within HA itself and provide an interface between external devices and apps (like add-ons but not just add-ons).
without an integration HA wouldn’t be able to interact with those devices or apps.
an example of this is zwavejs.
there is a zwavejs app (either an add-on or stand-alone docker container or a server that is installed on the host directly) that runs outside of HA. it is completely independent of HA itself (except in the case of the add-on it is controlled by the supervisor - which is also technically outside of HA itself).
then there is the zwavejs integration that provides the interface between the zwavejs app (via the add-on or etc) and HA.
in your case your vantagepro2mqtt add-on provides the interface to your weather station device and converts it to a protocol that HA can use but only by using the MQTT integration to set up that communication.
but you also need to run an MQTT server somewhere too. Which can be an add-on, a docker container or installed directly on the host and again is external to HA.
to interface between the MQTT server (created by the add-on etc) you need to use the MQTT integration which is built-in to HA.
HACS provides a way to add additional (or custom) integrations to HA as if those integrations were a part of HA.
The add-on store provides add-ons (community provided or officially provided by the HA team).
HACS provides integrations (or frontend components)
The way you add the two is also completely different.
integrations are activated from within HA in the devices and services section of the settings menu.
add-ons are installed and configured by the supervisor.
But I will say that the new menu layout does make it harder for the users who don’t understand how this all works to differentiate the two.
But also the headings at the top of the two pages when you finally get to the point of installing them does differentiate between the two pretty effectively. I mean, there is “add-ons” and “integrations” right at the top of the respective pages.