It's another one of those dastardly terminology issues - I think woodworkers are particularly prone to calling different things the same thing, and the same thing different things depending on who you ask!
I would disagree with CharlieHorse's answer and call the top pictured saw a handheld jigsaw, and the bottom pictured saw a sabre saw or reciprocating saw. Again in reality they are both types of reciprocating saw, and the term "jigsaw" can also refer to a fixed/bench-mounted version of the same thing as the handheld one shown above.
For my own definition I would say that a jigsaw has a shallow blade, and is meant for doing fine, controlled/intricate work generally to thin boards (like a jigsaw puzzle!) whereas a sabre-saw has a deeper blade and is particularly suited for doing fast but rough cuts and is of particular use for demolition when removing old windows or other timbers. Both sabre-saws and jigsaws can also be fitted with metal-cutting and other types of blades.
A further distinction is that what I call a handheld jigsaw is always meant to be used with the large flat base guide on flat material, whereas with the sabre saw it's often used as a quick "cut-off" saw without relying much on the guide, which is usually smaller and may be mounted non-rigidly so that you can easily follow contoured material.
Wikipedia currently claims that the top pictured saw may also be called a sabre saw, but I've personally never heard of it being referred to anything other than a (handheld) jigsaw. To be clear, "reciprocating saw" would be (in my mind) the general term for any type of saw where the blade moves back and forth.
Edit: I've done some research on Makita's, Ryobi's, Black and Decker's, and Bosch's websites (all large power tool manufacturers:
So they would tend to agree that the top is a jigsaw and the other one is a reciprocating saw, though I'd still contend that a "reciprocating saw" is a valid term for any kind of saw where the blade moves back and forth.