The first time I heard "roguelike" used to describe something, it was in reference to Dwarf Fortress's Adventure Mode (which, from what I've read, seems to have a lot more in common with actual classic roguelikes than a lot of modern games described as roguelikes do). At the time, I had no idea what "Rogue" was. Then, somewhere between 2012 and 2013, I found out about Spelunky (which was described as a platformer with roguelike elements*) and Rogue Legacy (which the creators called a "rogue-lite", presumably because it has some of the gameplay elements of Rogue [randomly generated levels, permanent death of individual characters] but by no means all of them).
Knowing what I know now, I would probably consider a "rogue-like" to be any game that has most, if not all, of the same sort of gameplay elements as Rogue, while a "rogue-lite" would be any game that has permanent death of your character (no extra lives), randomly generated levels, and a fairly high level of difficulty. In that sense, I guess "rogue-lite" is more of a genre modifier than a genre in and of itself.
*The creator of Spelunky, Derek Yu, goes into quite a bit more detail in his book "Spelunky", which you can get from Boss Fight Books. I highly recommend the book. In short, he says that he took elements from platformers that he liked (the sort of general gameplay mechanics platformers have), the elements from Roguelikes that he liked (permanent death, randomly generated levels, and having every kind of object in the game world be subject to the same consistent set of rules), and made a game** out of them.
**The original Spelunky, which was a pseudo-retro pixel platformer (it had pixel art and chip-tune music, and ran at 30fps). The version released on XBLA and Steam (Spelunky HD) came later.