The answer to this question is a resounding ... Sometimes.
Basically, there are a few things we need to clarify before we can respond to this properly.
What do you want to do?
The need for an audio interface depends entirely on what you want to do and how you want to do it. It allows you to use professional grade screens. Whether you need the level of sophistication, it will largely depend on the task at hand.
Compose - You do not need an audio interface to compose music. You do not need additional ins or outs. You can save expenses at this time in the process.
Mixture - This enters a gray area. If you use studio screens, you can decide to use an audio interface for a cleaner, true audio mix. You can benefit from using an audio interface, but it is not necessary.
Mastering - Here I will say yes, an audio interface is your best friend when you mastering. You need to listen in detailed details that will only be possible when using an audio interface.
What type of music are we talking about?
Is it just you playing the guitar? Do you work with a hip hop song? Do you want to record a friend who sings the latest song she wrote? You do not need multiple inputs from an audio interface.
A couple of people ... a trio? Maybe we want one, but no big scale is needed.
Want to record a full rock band? Lets think big ... how about an orchestra? You will need extra capacity for a significant audio interface.
As illustrated above, some genres of music require enhanced features, and you want an interface to help with clarity. Other genres can be quite simple to record and enjoy without all the bells and whistles.
What level do you work with?
Do you enjoy music at a hobby level? Do you like to get together with a couple of friends and a disturbance? Perhaps you just like sampling beats and plays with music. You probably do not need an interface at this level.
Do you want to record your band exercises before a show or recording? You may want to call some help from an audio interface, but its not that much. Its still not pro level things.
Do you start a music studio where you intend to record and produce all kinds of bands? Do you want to be prepared for anything in the door? Have an audio interface and know how to use it. For commercial reasons like these you should probably go with an advanced interface.
How much do you spend?
If youre the kind of person who always needs the best equipment, youve probably already realized that you need an audio interface ... You might have decided to spend top dollars for it.
If you still try to decide, the driver here is simple. How many inputs / outputs do you want? The more efforts you want, the more you will pay. You can get something easy for about $ 100 - $ 200, or go as wild as you like.
The general consensus is that you should keep your audio interface as simple as it needs while investing more in your monitors themselves.
Will you notice a big difference in the sound?
It really depends on the sound card in your computer. There has been a big improvement in sound cards in recent years. A high-end sound card vs. a low audio interface will probably provide the same sound quality, all the same.
On the other hand, if your sound card is not so good, and you go with an advanced interface, and advanced studio screens, you will undoubtedly notice a wonderful difference.
Does this help?
I know youre probably looking for a yes or no answer, but its definitely more of an essay question. You can find people who are on both sides of the argument, but I think most scenarios fall into the middle somewhere.
The audio equipment is expensive so its good to ask the question and think about what to do before you enter and spend a lot of money.
I hope this has given you some food for the thought. Have fun on your musical journey ... with or without the interface!