Why are there two separate branches of calculus?

Calculus just means a way of calculating. There are many types of calculus, some of which are modern and currently actively studied (e.g. functional calculus, umbral calculus, difference calculus), while some are more or less fully understood. Differential and Integral calculus, i.e. the standard "calculus sequence", falls in the latter category. Note: mathematical analysis is a very active field of study, but it consists of generalizations and applications of the calculus sequence.

There are also abandoned types of calculus, like techniques for calculating square roots by hand example, which used to be taught like long division in school.

It depends on what you're referring to when you use the term "calculus".

Calculus itself comes from the Latin word meaning small pebble used for calculation, and is used to not only refer to differential and integral calculus as you describe. There are many other fields that have the name "calculus" to describe methods of calculation, such as propositional calculus, lambda calculus, etc. Before "standard" calculus, the term was widely used to refer to any field of mathematics.

But in the scope of differential and integral calculus, you are correct; the idea of infinitesimal change and infinitesimal summation predate Newton and Leibniz by centuries. The Fundamental Theorem of Calculus was only developed in the 17th and 18th centuries to connect these ideas. First introduced by Leibniz's and Newton's precursors, it was refined by these two. "Standard" calculus was then rigorized by later figures with the definition of limits such as used in derivatives, contrary to "infinitesimal ratios" presented by Leibniz.