The "leftovers" are the unfinished part of a meal which you can keep in the fridge or freezer for another day. "The remnants" of something are small traces or pieces of evidence that indicate what the thing was, perhaps after consumption or destruction. If you only had the remnants of a meal, you might only have a few small pieces of meat or vegetables remaining - not enough to keep for another day. So "remnants" and "leftovers" are not synonyms in the context of meals.

They mean the same thing, but it's "leftovers" (not "leftover"), and "leftovers" is much more common. "Remnant" is more formal.

"Leftovers" almost always refers to food. Specifically, it refers to uneaten food, at the end of a meal, that can be saved and eaten the next day. "My wife always buys a big turkey for Thanksgiving. We eat leftovers for days, which is fine because I actually love leftover turkey." "Remnants" can refer to almost anything, not just food. It tends to mean small, unusable scraps. It carries a sense of "very little." Cloth for making clothing comes in long rolls called "bolts." After using it, there is usually a small piece left at the end that is too big to waste, but too small make a piece a clothing. That piece of cloth would be a "remnant." In the Aeneid, many Trojans were killed by the Greeks, and Aeneas, who survived, calls himself "a remnant left by the Greeks."