This is my 2 cents, though I'm not an expert:
"Probability Measure" is used in the context of a more precise, math theoretical, context. Kolmogorov in the year 1933 laid down some mathematical constructs to help better understand and handle probabilities from a mathematically rigid point of view. In a nutshell - he defined a "Probability Space" which consists of a set of events, a (σ)-algebra/field on that set (≈ all the different ways you can subset that original set), and a measure which maps these subsets to a number that measures them. This became the standard way of understanding probability. This framework is important because once you start thinking about probability the way mathematicians do, you encounter all kind of edge cases and problems - which the framework can help you define or avoid.
So, I would say that people who use "Probability Measure" are either involved with deep probability issues, or are simply more math oriented by their education.
Note that a "Probability Space" precedes a "Random Variable" (also known as a "Measurable Function") - which is defined to be a function from the original space to measurable space, often real-valued. I'm not sure, but I think the main point here, is that this allows us to use more "number-oriented" math, than "space-oriented" math. We map the "space" into numbers, and now we can work more easily with it. (There's nothing to prevent us to start with a "number space", e.g., $\mathbb R$ and define the identity mapping as the Random Variable; But a lot of events are not intrinsically numbers - think of Heads or Tails, and the mapping of them into numbers 0 or 1).
Once we are in the realm of numbers (real line R), we can define Probability Functions to help us characterize the behavior of these fantastic probability beasts. The main function is called the "Cumulative Distribution Function" (CDF) - it exists for all valid probability spaces and for all valid random variables, and it completely defines the behavior of the beast (unlike, say, the mean of a random variable, or the variance: you can have different probability beasts with the same mean or the same variance, and even both). It keeps tracks on how much the probability measure is distributed across the real line.
If the random variable mapping is continuous, you will also have a Probability Density Function (PDF), if it's discrete you will have a Probability Mass Function (PMF). If it's mixed, it's complicated.
I think "Probability Distribution" might mean either of these things, but I think most often it will be used in less mathematically precise as it's sort of an umbrella term - it can refer to the distribution of measure on the original space, or the distribution of measure on the real line, characterized by the CDF or PDF/PMF.
Usually, if there's no need to go deep into the math, people will stay on the level of "probability function" or "probability distribution". Though some will venture to the realms of "probability measure" without real justification except the need to be absolutely mathematically precise.