What’s the importance of an audio interface?

You need it to plug your mic or line into to record to your PC DAW. You need it to listen to music on stereo monitors from your DAW. You need it to export songs from your DAW.
 
When it comes to recording the difference is worlds apart. Other than that it makes producing a little easier and it takes load off the CPU.
 
Your audio interface is primarily there to convert your analog sounds to digital and vice versa. It is the A/D or D/A conversion algorithm that is important when choosing your interface. The higher quality interfaces have better (arguably of course) INTERNAL A/D and D/A convertors. A/D means Anal to Dig; D/A means the opposite.

As an audio interface, you're concerned with the A/D convertors. How well will I be able to convert my analog sound (guitar, mic, kazoo) into a digital file that my DAW can use to let me edit? Your interface does that. It also processes it much faster than your computer's basic sound card can do and you improve the 'latency' time when you are trying to hear the same sound you're playing get processed and back out to your speakers.

You don't need to spend a ton of money on an interface but not having a dedicated interface is not a good option. The other final point is that an electric guitar requires a Hi-Z level input, a mic requires a mic-level input (with or without phantom power) and an external device requires a line level input. Most interfaces now take care of handling all three levels of inputs. That's the other benefit.
 
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