Strictly speaking, Saria's Song is the melody you play on the ocarina in the game, while Lost Woods is the background music you hear in…well, the Lost Woods.
All of the non-warp songs in Ocarina of Time consist of a three-note motif that's played twice, and then an "answering" melody. For example, here's Saria's Song.
This is what Link plays when you press C-down C-right C-left C-down C-right C-left on the ocarina. As you can see, it's only a couple of bars long.
But, all the non-warp songs are also extended into full pieces used as background music somewhere in the game. For Saria's Song, the extended version is the background music in the Lost Woods.
So, to be pedantic, Saria's Song is the short ocarina melody, and Lost Woods is the full, harmonized version. This is how you'll find them labelled on the OST album: Saria's Song is track 34 and Lost Woods is track 35. (Similarly Song of Storms versus Windmill Hut, Epona's Song versus Lon Lon Ranch, and so on.)
But, if you're doing a cover, there's not much point in covering the ocarina version. It's only three bars long! So outside of the OST, most people use the names interchangeably. If you find a cover of "Saria's Song" somewhere, it's almost certainly actually a cover of Lost Woods. And since nobody really covers just the short ocarina melodies, in practice, this doesn't cause much of a problem. It just makes it a little bit harder to search for tabs.
Sheet music taken from here