Computers usually report time in UTC but don't actually have leap seconds. Most computers are pretty bad at keeping time, and regularly sync to NTP servers for the current time. NTP servers generally have better time keeping hardware (or sync with better sources). When a leap second occurs, NTP servers will smear that second over 12 hours or so. Different NPT servers have different smear standards. During that smear, seconds are a bit longer, or shorter for negative leap seconds, than usual. But for general computing this doesn't matter. The smear is on a similar order of magnitude of normal clock drift. This isolates the leap second to only computers that need to track it. UTC/UT1 is designed to be used by humans and is based on the Earth's revolution and rotation about the sun (well now it's actually measured using distant pulsars).
The big reason to isolate this is because leap seconds aren't totally determinate. We can predict them to some degree, but ultimately rely on measurement. Unlike something like leap days, you cannot safely code a system that accounts for future leap seconds. Leap days follow a well-defined formula, leap seconds do not.
For time sensitive applications like navigational computers, TAI is used. TAI is currently 37 seconds off from UTC. TAI is associated with the SI Second unit. While originally derived from the solar day, it is now based on the vibrational frequency of a caesium atom.
For a more detailed explanation see http://mperdikeas.github.io/utc-vs-ut1-time.html for a good summary.
For an overview on how Google handles the smear see https://developers.google.com/time/smear