What is the difference between using a jewel with "increased fire damage" instead of one with "increased burning damage"? Which one has the largest effect on the damage I deal?
There is no difference. Use the one with the higher number, or the one which has other helpful modifiers.
All sources of Increased Damage which apply to the type of damage you are dealing are added together when calculating damage. There is no benefit to taking bonuses with multiple different sources of Increased Damage -- they all have the same effect; they just apply to different things.
Path of Exile uses the term "burning damage" as a synonym for "fire damage over time". It does not have any special meaning beyond this. In practical terms, this primarily describes damage done by igniting an enemy (which has a chance to occur when you hit them with fire damage); by the skills Righteous Fire, Scorching Ray, or Searing Bond; or by burning ground (e.g, the trails left by Flame Dash or various other skills).