Data analysis usefully defined, via wikipedia, emphasis mine:
Data analysis, also known as analysis of data or data analytics, is a process of inspecting, cleansing, transforming, and modeling data with the goal of discovering useful information, suggesting conclusions, and supporting decision-making.
Meaning, the focus is on deriving information, insight, or conclusions so that humans may do, understand or decide better.
Tom Mitchell wrote a very helpful definition of in his book Machine Learning, as quoted on wikipedia:
A computer program is said to learn from experience E with respect to some class of tasks T and performance measure P if its performance at tasks in T, as measured by P, improves with experience E.
Here, the emphasis is on programming agents that learn to do things—predict housing prices, transcribe text—from experience, i.e. data.
This, I find, is a helpful distinction: Instead of method, we can reason about goals. If we're analyzing Fourier coefficients of recorded speech to understand how the human brain parses sounds into vowels, that's analysis; if we're doing to write a program that automatically transcribes text, it's machine learning.
(Very diverse field, little consensus re terms, your mileage may vary.)