Sometimes I see people struggling with the difference between these two functions (usually the infamous INxPs). Both are subjective rational functions, but their orientation is different. Fi seeks to come to a subjective conclusion (subjective-subjective) but Ti seeks to come to an objective conclusion (subjective-objective).
I was reading about Virgina Woolf's To the Lighthouse (which, TBH, kinda bored me even with the fantastic writing), and the artist in the story at the conclusion betrays a Fi mindset (I suspect Woolf herself to be INFP as well). To quote wikipedia: "...she realizes that the execution of her vision is more important to her than the idea of leaving some sort of legacy in her work."
In short, Fi seeks to come to a conclusion/understanding that the person agrees with personally. It is congruent with their ideals and sense of themselves. The "right" answer. Ti, on the other hand, seeks to come to a conclusion/understanding that is accurate. It is also "right", but in a different sense. The end result may be something the Ti-user finds disagreeable, but it is what it is. The conclusion has to stand separate from the person, much like the "legacy" in the quote above. Ideals and personal wishes are irrelevant.