I think you're pretty confused.
AT&T is NOT a plan. AT&T is a telecommunications company that provides many different types of services and equipment for various plans/services. AT&T sells home Internet access (DSL, U-Verse, and/or fiber) and cell phone service among other things.
Based on your question, I'm assuming that you currently have an AT&T cell phone plan and/or mobile data plan. You have to understand that cellular service relies on a good and strong connection to the cell tower. If there are too many people on that one tower, things will slow down for everyone. Additionally, if you go over your data limit, pretty much all carriers will slow you down to a crawl until the next billing cycle.
Without any specific information, like hotspot model, data plan, etc, no one here can "magically" know what's going on.
A router and modem are two very different things that can't really be "compared" the way you're asking. A router is a device that routes (hence the name) traffic between networks. A modem is a device that 'translates' the signal coming from your ISP (through the phone line), into something the router can understand and work with. However, generally speaking, if you get an Internet plan with AT&T, they will sell/rent you a combo unit which includes a router, modem, and Wi-Fi AP in one unit.
If you want more specific help, you're going to need to be more specific.