The main difference is in the price and the quality. The fender guitars are more expensive and tend to be of higher quality, whereas Squier guitars are made more cheaply and have lower quality. By this I don't mean that the Squier guitars are low quality and bad, but (usually) they are not so good as a Fender guitar.
By good and low quality, I refer mostly to the sound, the playability and the durability of the guitar. A high quality guitar will be able to produce a high quality sound, whereas a low quality guitar's sound will not. Similarly, a low quality guitar might not be that easy to play. Using low quality woods might cause pain on your arm, whereas a high quality guitar will have woods that are comfortable for your hands, so you can play for hours without getting any pain. Lastly, a low quality guitar isn't made to last a lifetime.1
The Squier guitars are being made in China and Indonesia, but in the 80's they manufactured them in Japan and they were of pretty good quality. If you look for used squier guitars from the 80's (on Ebay or something) you'll see that they are more expensive than some new ones, and that they where made in Japan.
The Fender guitars are made in the USA and receive more attention in the production. Some of the guitar parts might be of higher quality (not necessarily all of them), but what makes the quality better is the whole production.
You should also look into the Mexican Fender Guitars. They are somewhat above the Squier models and somewhat below the American ones, both pricewise and qualitywise. Again, not every model will be like this.
If you have time, you go in depth in this article: Squier vs Fender Stratocaster Guitar Review. It shows many different squier and fender models and it explains what's up with everyone.
1Keep in mind, that a beginner won't easily notice all these differences. That's why many people start off with these kind of guitars (like me). They are great for someone who doesn't want/cannot spend a lot of money on guitars just yet, so a cheap solution is a good choice in this case.