Cater, purvey, pander are comparable when they mean to furnish with what satisfies the appetite or desires.
Cater basically implies the provision of what is needed in the way of food and drink. The term especially implies provision of food and drink ready for the table. In extended use cater often implies the provision of something that appeals to a specific appetite.
Often, especially when followed by to, the term implies a certain subserviency (as to popular standards or uncultivated tastes).
Purvey usually suggests the provision of food but sometimes of such other material necessities as lodgings and clothes. In contrast with cater, however, it suggests service as a source of supply, either as an agent through whom what is wanted may be found or as a merchant who sells the needed articles. In extended use, especially when followed by for, purvey implies the provision of whatever is needed to satisfy, delight, or indulge.
Pander, which basically means to act as a procurer or as a go-between in an illicit amour, in its frequent extended use may imply a purveying of something which will gratify desires and passions that are degrading or base or may connote mere servile truckling or even no more than a deferring to or a reasonable indulgence of tastes.