This is an old post, but this is a topic I know a lot about. I'm a grad student in math and spend a lot of time reading about the history of math, specifically early calculus. When it comes to the Newton vs. Leibnitz debate, the short answer is that isn't not really a "toss-up" and more so that both men deserve to be credited for the same thing. For example, if two people independently invented something like the can opener, it's fair to say that both people are the inventors of the can opener. In math, it's not atypical for something to be invented multiple times independently like this. For example, the concepts of a number system, pi, the Pythagorean theorem, etc., were all re-invented repeatedly throughout the world. In fact, before calculus, both Fermat and Descartes independently invented analytic geometry (basically stuff relating to graphing functions and curves).
Now when it comes to calculus specifically, we have to be very precise with what we mean by "inventing calculus," because at what point is something considered "calculus"? The term calculus means "small pebble" in Latin, referring to the general idea of looking at how things change on an arbitrarily small level. It would make sense to just define "calculus" as anything falling under this category, so we should describe "inventing calculus" as just inventing any math falling under this umbrella, right? But this leads to a weird issue, which is that mathematicians have been messing with this idea for centuries. The oldest I can think of would be the Pythagoreans accidentally inventing the sqrt(2), dating back to roughly 500 BCE, over 2,000 years before Newton or Leibniz. Instead, we generally choose to describe someone as the "inventor of calculus" if they have met three criteria:
If someone did all three of these things independently, they are considered to have invented calculus. The reason we choose this is because there was already a pressing issue at the time to figure out how to properly describe a curve changing over time if we can't simply find the slope. If we could do this, and then essentially "undo" this, that would be quite handy! So let's look at both Newton and Leibniz and see how they meet this criteria.
For Newton, he had already discovered physics and was working on calculus to describe the motion of an object using math. Throughout Newton's work, a lot of it focuses on physics, which is why a lot of his supporters at the time were physicists. Newton referred to derivatives as "fluxions" and you could think of "fluents" as integrals. One of Newton's main discoveries was how to managed to break down a curve into an infinite polynomial. The benefit to this is that instead of having to look at weird functions like sin(x), 1/x, etc., you just have to focus on functions like ax + bx^(2) + cx^(3) + .... This allowed Newton to find a relatively simple algorithm to find the derivative. Newton even describes in De analysi how to properly use the inverse of this algorithm to find the area under a curve, heavily implying the use of the Fundamental Theorem of calculus. He found a method for find the derivative of a curve, a method for finding the area under a curve (integrals), and clearly recognized that one was the opposite of the other. Therefore, we can say Newton invented calculus, according to our definition from earlier.
Now let's look at Leibniz. Leibniz was not a physicists, so his work focuses on calculus from a more pure mathematical point of view. He treated calculus more like geometry instead of physics, which is why most of his supporters at the time were mathematicians. Like Newton, Leibniz also starts off focusing on infinite series and sums, though he does not dive so much into polynomials. In fact, the sign for integration comes from Leibniz, as he simply considered integrals as an infinite sum of rectangles under a curve (which is why the sign is a long S, for sum). Meanwhile derivatives were referred to as "differentials" or "differential quotients," referring to the difference of two points across an "infinitely small" distance. It starts to become more clear how if Leibniz viewed integration as adding up several small things, while derivatives were subtracting small things, Leibniz would clearly see on his own that one would be the opposite of the other. And in fact, the notation for a derivative as dy/dx comes from Leibniz because dy simply referred to the difference in height of two "infinitely thin" rectangles and dx was the thickness of those rectangles, like so. The sum of these rectangles is just integral. This is how Leibniz came to understand the Fundamental Theorem of Calculus. Therefore we can say Leibniz also invented calculus by our definition.
Now this doesn't address a key issue, which is the fact that both of these men were in mid-1600's Europe at a time where they could easily communicate with each other and discover each others' works, so how are we sure they actually independently came up with these ideas and didn't steal them from one another? This was actually quite the debacle at the time and would warrant its own post entirely unfortunately. To sum it up briefly, Newton tried to publish his work in the 1670's, but could not publish it for a few more decades. In this time, Leibniz published his work, so both men were not aware of the other's work while they invented calculus and managed to independently create a lot of the same ideas. There were several times that both Leibniz and Newton were accused of stealing from the other, but there has never been evidence to support this, and it always came from supporters of either person (e.g. Johann Bernoulli, a student of Leibniz, frequently accused Newton of stealing from Leibniz and despised Newton).
I should also note that this is not the only way to describe "inventing calculus," but other interpretations lead to several others being referred to as the "inventor of calculus." As mentioned earlier, you could say the Pythagoreans invented calculus. You could also say that it must be when limits were properly defined, which would mean Cauchy invented calculus about 100 years after Newton and Leibniz. There are also more obscure arguments about how Barrow or Fermat invented calculus because they had more specific ways of vaguely describing limits, but none of these arguments are typically considered as strong as the definition given earlier with the Fundamental Theorem of Calculus. That is why we typically consider just Newton and Leibniz as the inventors of calculus.
Sources:
Mathematics and Its History 2nd Edition by John Stillwell
From the Calculus to Set Theory 1630-1910 by I. Grattan-Guinness
Who Gave You the Epsilon? Cauchy and the Origins of Rigorous Calculus by Judith V. Grabiner
