Chess Engines and Humans: The Difference
It's safe to say that there is a huge difference between how humans and chess engines play the game. Not only do they have around 800 more Elo than the best human players in the world, but they can calculate billions of moves every second. That is more than you or I will probably ever see in our lives.
So the question might come to mind: What is the difference between engines and humans? What is the distinction between how we see the board, our playing styles, how we calculate, etc? If you are curious as to how chess engines actually work, check out my article on them here: How Do Chess Engines Work?
Human vs. Engine: Playing Style
All engines are incredible at calculating very deeply and accurately. An engine's playing style is just whatever they deem is best, they have no biases towards certain moves.
Humans certaintly have styles, because humans are in nature biased, therefore certain moves will stand out more than others. For example, Tigran Petrosian was a very cautious and defensive player who tried to limit the threats his opponent could make. His nickname was "Iron Tigran" because of his seemingly impenetrable defense. Another legend like Mikhail Tal had a creative and attacking style. His brilliant sacrifices showed his love of taking risks, the complete opposite of a player like Petrosian.
Human vs. Engine: Calculation
Engines use game trees to calculate. A game tree are branches of every possible move, every move from each move, etc. It is very similar to a family tree, except instead of genetics it's chess moves. For example, in one chess position, there are 5 possible moves, 2 possible moves from each of the first moves, and 1 or 2 moves from there. The tree will look like:

Because they look for each possible move through game trees, they can determine the most advantageous line for whatever side they are playing. Engines will always assume that their opponent will play the "best" move, and base their calculation off that.
Humans have the same assumption, but can miss moves due to the ambiguity of them or lazy calculation. Engines can miss moves as well, through a process called pruning. Pruning is cutting off branches or variations from the game tree that they deem to be bad for whatever side is playing to limit how much they have to calculate. In extremely complex situations that require imaginative thought such as the position below, it takes the engine longer than usual to find the mate in 5 for White. Can you see it?
Humans think very differently. A human will not calculate every legal move for 20 moves like a computer. Instead, when a person looks at a position, they have certain moves that stand out. For instance, in the position below, a human will most likely not calculate a move like Ba6 because that just hangs a bishop for no compensation. An engine will calculate that move and every other blunder, because they simply don't know if it's a blunder or brilliant. A person will look at the position below and think about moves such as Nc3 or exd5, because a human knows that hanging your bishop for nothing is probably bad only 3 moves out of the opening.
It is important to understand that if a human was capable of calculating billions of moves every second like an engine can, they would be just as good as their titanium counterparts. But people are not capable of this function, and will therefore never be able to reach the playing ability of an engine.
Human vs. Engine: Viewing the Board
Engines view the chess board as a series of numbers. For example, an empty square may have a value of 0, a White pawn 1, a White knight 2, etc. It will then use a bitboard to represent that. Bitboards are data arrays, with a number represented each square or piece on a board. In chess, the bitboards are 8x8, which makes 64 bits. For example, in the position below, the computer will see:
Humans see the board, well, like how you see the board. Each square is a square, not a number, and each piece has a unique shape, not value.
Conclusion
In conclusion, engines and humans think very differently about all aspects of the game. Perhaps our only similarity is how the pieces move! Even so, we may never be able to reach the level our coded rivals are at, but we can certainly learn from them and recognize our differences. We should not try to brute force calculate all that they can, but to understand the game well enough to compete with our flesh companions.