One main difference is that synthesis involves multiple sources, while analysis often focuses on one source.
Conceptually, it can be helpful to think about synthesis existing at both the local (or paragraph) level and the global (or paper) level.
Global synthesis occurs at the paper (or, sometimes, section) level when writers connect ideas across paragraphs or sections to create a new narrative whole.
A literature review, which can either stand alone or be a section/chapter within a capstone, is a common example of a place where global synthesis is necessary.
When you are reading a good story you will typically take in the information you are reading about and create in your mind a picture of what you are considering (there are other types of synthesis in literature, such as forming opinions, generating ideas, etc., but for this illustration, we'll stick with forming images in one's mind).
For example, I recently finished reading the second book in The Hunger Games trilogy, entitled Catching Fire, by Suzanne Collins.
As I read the book, my natural inclination was to consider things, such as the way the characters described their surroundings and how they reacted to them, and pair that with my own knowledge of shapes, landscapes, types of plants and animals, etc.
I did this to come up with a picture in my mind of how the game arena might have appeared.
It might be Catching Fire as previously mentioned, or perhaps something completely different.
After reading it, you feel compelled to write to the author and tell them how much you thoroughly enjoyed their book.