Say that a character has 5 HP remaining and is dealt 10 damage from an attack. Of course, 5 - 10 = -5, so the character has dropped below 0 Hit Points and follows the rules for making death saving throws (provided they didn't get dealt enough damage for instant death). However, it's not clear to us if the character remains at -5 Hit Points or if they bounce back up to 0 Hit Points (like many rules in 5e, knowledge of past editions may be a hindrance to interpreting them).
If negative Hit Points exist, then characters will take longer to recover naturally (since stable characters recover at a rate of 1 HP per 1d4 hours). Also, a natural 20 on a death saving throw, which recovers one Hit Point, would not make them instantly conscious. Finally, it would mean that instant death is a greater possibility, as you need less to reach the threshold if you are attacked again.
However if negative Hit Points do not exist and characters bounce up to 0 after crossing the 0 HP threshold, then characters will always regain 1 HP and become conscious after waiting 1d4 hours or rolling a natural 20 on a death saving throw. Also, this would mean that instant death is far less likely because someone who attacks an unconscious character would always need to deal maximum HP damage or they don't kill you (and if they don't, then I guess their damage means nothing, which seems rather odd).
Unfortunately, the example provided with the basic rules isn't helpful because it describes someone taking enough damage for instant death, but not someone who got less than that. Furthermore, the rules describe "Dropping to 0 Hit Points", but not "Dropping Below 0 Hit Points" and seems to omit what happens when you take more damage than the HP you have remaining, but less than enough for Instant Death. Our group spent a while debating this when we played from the Starter Edition and we weren't sure given that some previous versions of D&D had them while others didn't. So do negative hit points exist or do characters "bounce up to 0 HP"?