I'm not saying these and other business phrases aren't widely misused or used as a way to sound smart while saying nothing. But in case anyone is interested:
Synergy
The term comes from M&A and refers to scenarios where the whole (after a merger or acquisition) is worth more than the sum of the parts. For example, a steel foundry which acquires a competitor and can now buy iron at a lower price because it buys in higher volume. The increase in profitability of both foundries from the decreased cost of iron is a "synergy".
Value-add
Comes from Michael Porter (I think) and refers to the idea that each of the activities a company performs on their inputs before the final output should add value to the final product. If the company is unable to add value through one of those activities it's something they should pass to another party. This leads to phrases like "is our customer service call centre a value-add or should we outsource it?"
My beef with things like this is not with the concept, but the way it is used. 'Does our call center add value to our product?' is an entirely reasonable question. 'Value-add' as a noun is a big red flag for me, suggesting the speaker has a poor grasp of the underlying concept and/or is trying to overstate the significance of the issue to seem more important. I feel the same when someone uses 'dialog' as a verb.
Your definition is good enough, but "synergy" actually comes from Buckminster Fuller who built a philosophical system mostly around the idea in the 1950s through early 1970s. The business community got it from him, largely, from what I heard, by way of a bunch of leftish techies that were associated with him and Black Mountain College in the sixties.
Uh, according to Wikipedia the word comes from Greek (like so many other in English), 'synergia συνεργία from synergos, συνεργός, meaning "working together"'. They do mention that Buckminster Fuller analysed it further, and coined "synergetics". See http://en.wikipedia.org/wiki/Synergy.
Oh, and to the grandparent (I think), it was fun to see "M&A" used in a thread about opaque language. Turns out it stands for "mergers and acquisitions" (http://en.wikipedia.org/wiki/M%26A).
Synergy
The term comes from M&A and refers to scenarios where the whole (after a merger or acquisition) is worth more than the sum of the parts. For example, a steel foundry which acquires a competitor and can now buy iron at a lower price because it buys in higher volume. The increase in profitability of both foundries from the decreased cost of iron is a "synergy".
Value-add
Comes from Michael Porter (I think) and refers to the idea that each of the activities a company performs on their inputs before the final output should add value to the final product. If the company is unable to add value through one of those activities it's something they should pass to another party. This leads to phrases like "is our customer service call centre a value-add or should we outsource it?"