A summary is a short description of your book; think if it as a sales pitch or the blurb on the back of the book. It's 1-2 paragraphs, no longer than half a page. You hit the hook, the teasers, the main conflict. Ideally, it's what you would put in your query letter.
At a conference a few years ago, I heard an agent describe it as what you would say if you suddenly found yourself on an elevator with an agent or editor, who turns to you and asks, "What's your current work-in-progress?" You have until the end of the elevator ride to get them hooked.
A synopsis is longer and can be up to 2 pages. It's more like an abbreviated Cliffs Notes for your book. Write it in third person (even if your book is in first person), present tense, include your main characters, their motivations, conflict, major plot events, setting, themes, AND the resolution. (That means, if it's a murder mystery, you tell who dun it.)
The synopsis should be representative of your writing skill, so make it shine. It shouldn't read like a user manual or a dry encyclopedia entry. Punch it up with the same sensory based imagery, tone, and humor that occurs in your book.
As to which I would rather see? The answer is BOTH. This is particularly true for a values based publisher, like here in the LDS market. Example: A novel about a teenage coming of age story. From the summary, I might be interested. But a synopsis would tell me that in chapter 17, she discovers she's pregnant and decides to have an abortion. That just wouldn't fly in my market and I'd like to know that before I'm 150 pages into the manuscript.
Careers taking off at a writer's conference? It could happen. I know several people who've gotten that toe in the door from a conference—submitting to an agent that spoke at the conference, winning an award at the conference which got them a bypass the slush pile free card, and signing up for a actual pitch sessions. Readers—do you have any success stories you want to share?