Three broad reasons for binary research. Rounding errors, research efficiency, and flexibility.
Rounding errors: fractional beaker or gold is rounded down at the start of each turn. If you're generating 15 commerce per turn, and splitting it 70/30 on research/gold, you're getting 10 research and 4 gold per turn; you've "lost" 1 commerce per turn to rounding. This can be a big deal early in the game, although the impact is negligible later on.
Research efficiency: every civilization you know who has discovered a tech will give you a small multiplier boost to your own research on that tech. When you frontload banking up the gold, until you have enough to finish the tech all in one swoop, and delay starting the actual beaker generation as long as possible, that increases the chance that neighbors will discover the tech before you start actually committing commerce to beakers, and your own research will be more efficient as a result.
Flexibility: If you spend 10 turns at 30% slider to research a tech, you're basically committed from turn 1. If you spend 7 turns at 0% research, you now can run 100% slider for three turns on any tech you want - you kept your options open. Or you could even spend that gold on something else, like upgrading units if an enemy just declared war on you. If, for example, your plan was to get Music for the free Great Artist, but after six turns at 0% slider you saw an opponent got Music, you could potentially pivot into some other tech instead. Binary research lets you make more informed choices on tech by delaying the point of commitment.
Edit: As for when you might do it, basically always if you aren't running at 100% research slider anyways or in the mop-up phase where the game is basically won and you aren't worried about being a little less efficient. It's just a good general habit to form.
your empire is generating. If it's an even number, 50% is as good as 0/100%.
/
due to rounding is just summing 
), and having a cash reserve from doing 0% slider for a while gives you the option of throwing money at the RNG if it decides to roll double snake eyes. The downside of this is that, post-Currency, an AI might demand your entire gold reserve (hi Joao 
) and start plotting if you refuse. That said I don't think there's too many situations where an AI would demand an outrageous sum of gold but wouldn't have demanded a high end monopoly tech instead, so don't let that discourage you from doing binary research. Good diplomacy skill will see you through most of those situations, anyway.
