Can the Arduino interface with USB devices without the USB host shield? I'm looking to connect various USB-devices to an Arduino Uno (specifically, a gamepad and a webcam, but I'm interested in the general case). Is this possible without using the USB host shield, since the Uno has a USB port and USB-to-serial converter? I could use the shield, but I'm curious if/why it's neccessary.

The Sparkfun website states that:


  The SparkFun USB Host Shield contains all of the digital logic and analog circuitry necessary to implement a full-speed USB peripheral/host controller with your Arduino.


Does the Arduino not already contain this, just needing the proper software (although obviously it's easier to use the already written library for the Host Shield). Or are there hardware limitations?

According to 
Can a Ethernet USB Dongle be instead of ethernet shield for arduino?
(Majenko's answer), it can, in theory, be done with the Due, but not with the Uno. In that case, is this because the Uno uses the ATmega16U2 instead of the FTDI FT232RL for USB communication?
First things first you need to learn the basics of how USB works.

In USB there are two main "things": Hosts and Devices.

You have one Host (usually your computer) and many Devices (such as Arduinos, Printers, Scanners, Webcams, WiFi dongles, etc).

Being a host involves knowing about what devices you are going to have attached to you and how to communicate with them. Being a device means just communicating with the host in your own way.

As you can see it is a far more complex job being a host than being a device.

Arduinos are devices. The small USB interface chips they use are very low powered and can only be a device, never a host.

If you want to plug a device into something it has to be a host (note: I am purposely ignoring hubs here). 

The Arduino Due uses a far more powerful chip, and that is capable of being a host (using what is known as On-The-Go mode).

To do it with anything less powerful than a Due you require additional hardware that is specifically designed to be a USB host - and that is where the USB host shield comes in. It has a chip that is designed specially to be a USB host so you can plug a device in to it.

None of the chips on the Uno are designed to be a host. They just don't have the power.