Thank you for such a great question! One of the great things about science is that there can be many different approaches to answering these types of big, meaningful questions. One way to answer a question like the one you ask is called evolutionary psychology, which I study at UC Santa Barbara.

Evolutionary psychology begins with the observation, like you do, that all animals are designed to maximize the survival and, most importantly, the reproduction of their genes. The designer in this case is the process of evolution by natural selection: genes that happen to have effects that make them better at replicating themselves in a given environment are the genes that tend to increase in number.

The process of natural selection, over geologic time, has sculpted the great variety of life we see all around, with each organism adapted to its particular environment. Homo sapiens are no different. Our species, and our brains and minds, have evolved to exploit the environment that our species inhabited for hundreds of thousands of years. Evolutionary psychologists have found evidence that, much like our heart is designed to pump blood or our eyes to see, our minds are designed to solve all sorts of other problems we faced in the distant past, like navigating, communicating with a spoken language, finding the ideal mate, finding the best food, maintaining good friendships, being treated fairly, helping your family, and many others. In this way, humans still do very much care about their survival and reproduction, even today.

But the brain/mind, you note, can do all sorts of things that it was not designed to do, like solving calculus problems, playing the stock market, and drafting legislation. Evolutionary psychologists argue that these evolutionarily novel abilities come about as a byproduct of our existing abilities. So the same nerves that control our natural ability to walk upright also give us the ability to surf, even though natural selection never designed our nerves for surfing. In all domains of modern life, it seems that we have to make do with a brain that was designed for the stone age. It just so happened that natural selection designed our brain to be capable of language and making lots of inferences, enabling the accumulation and expansion of knowledge over time into areas that have little to do with our direct survival and reproduction.

I hope this provides some of an answer to your great and big question.

I thought we had brains that have a bigger frontal cortex in the front of the brain, compared with other animals. But I'm finding links on google that disagree with this idea:

Gorillas Agree: Human Frontal Cortex is Nothing Special - Scientific ...

There's an idea that we became intelligent in special ways after we started cooking our food! We didn't need to use so much of our head for big teeth and jaws, for chewing raw food.

Here's an article about that, from the Smithsonian magazine:

What are the economy, politics, and appearance if not interest in survival and reproduction? Humans make more plans for the future than just about any other animal, but very few humans devote much effort to goals other than that.

There are humans who do art and science, but art has been observed in other animals, and science is just a sophisticated expression of curiosity, which is common among the more intelligent animals. Humans do some things more than other animals do, but there isn't anything that humans do that is truly unique.