Do you need a router if you have a modem?

That's like asking which is better a plate or a cup.

They are not interchangeable devices, they each have their own unique purpose. What makes it confusing is because you can get a modem-router combo.

A modem is a device that takes the cable medium (coax cable, phone, fiber, etc) from the IPS and converts it to ethernet (not just converting the cable but the entire data structure).

A router allows you to create a local network and share resources. It gives you private address to provide a much needed first layer of security, and then it routes traffic between your local network and the outside network (the internet from your ISP).

A modem-router is a box that has both built into one box. Typically this is a mediocre modem combined with a mediocre router. A $50 modem + a $75 router will be far superior to a $100 modem-router combo.

You need a modem if your ISP provides internet service over an analog carrier, such as cable TV or telephone lines. You don't need one if your connection is via optical fibre - you'd need an optical interface for your computer or router.
You need a router if you have a complex network, or multiple paths to the internet. Otherwise, strictly speaking, you just need a switch.
But for most home users using IPv4, they will only be allocated one address by an ISP. If they have more than one computer or networked device, they will most likely need a home router which uses DHCP to create local addresses for all devices in a private address space such as 192.168/16, and NAT to allow these devices to share a single public internet address. This router often includes a wifi access point integrated into the same box.

Personally, I have a modem to connect my PC to cable TV service, and use software on my PC to implement NAT, DHCP and DNS to connect a local network on a second interface. So in effect my PC is  acting as a router.