What are the advantages/disadvantages of using a popup password recovery vs a new page?
The difference is in the style of your product.
If you introduced popups in your product, you can go with popups, if not then try to avoid popups.
Advantages are in the favor of style.
Disadvantages: If you are using third-party add-ons as a handler for modals/pop-ups, this can cause you bad UX in terms of loading these modals/pop-ups.
EDIT: If your interactions are right, once again, it's all about style.