Is there a difference between "unity" and 1 in applied mathematics? Is there a difference between "unity" and 1 in applied mathematics? 
I know mathematicians have "roots of unity" and "partitions of unity",
but at least those have become standardized. In applications,
"unity" seems to be used randomly and maybe 
interchangeably with the number 1. Is there some subtle meaning
there that I'm missing?
 A: In my experience, "unity" is just a fancy word for the number "1". In the two examples you gave, that's what it means, certainly.
"Partition of 1" looks odd, to me. "Roots of one" risks confusion with other meanings of the word "one". So "unity" is useful, rather than merely ostentatious.
Any algebraic structure might have an element that has properties analogous to the number "1", but this thing is typically referred to as a "unit", not as "unity".
A: The OED's second entry for "unity" gives three different mathematical senses:

2.
a. Math. The abstract quantity representing the singularity of any single entity, regarded as the basis of all whole numbers; the number one.
b. Math. An instance of unity (sense 2a) as a mathematical object, esp. as a component of a number; an occurrence of the number one in a mathematical expression or calculation; (formerly also) †a whole number (obs.). Cf. unit n. 1a. Now rare.
c. A quantity, magnitude, or substance regarded as equivalent to the number one in calculation, measurement, or comparison. Cf. unit n. 1b, 10a.

A: In my limited experience (engineering and abstract algebra courses a couple of decades back) unity is not a synonym for 1, but a generalization of 1. It's the element of some set that leaves elements unchanged when a certain operation is applied - an identity element, and specifically the identity element for multiplication rather than addition, in contexts where both of those operations are defined. I've heard people say that some parameter or item is "unity" instead of "one" to emphasize its behavior as leaving things unchanged (or un-scaled), rather than being a particular quantity. Searching through some old algebra textbooks seems to confirm this usage.
