<---- deep, inside (subjective = to me)
Ti - right/wrong - literally it is logic
Fi - good/evil - literally it is emotions
I often use the term emotions for Fi as well, but I don't think it's a satisfactory term, because there is more to Fi than emotions, and more to emotions than Fi. In other words, Fi only relates to some aspects of emotion.
To explain an aspect of what Fi does, I'll contrast it with Ti.
Ti looks at something and divides it into its parts, analytically. Or, it may choose between 2 or more selections.
Fi on the other hand brings parts together, or matches things up, or finds a balance where 2 things can co-exist.
So, they do the direct opposite of each other in a way.
The way Fi relates to emotions is because as Fi goes on its way, it is "pinging" the environment off of an internal register whereby there is a note of whether the things are in accord or out of accord with that register. The way that register is experienced is such that it is somewhat suitable to call it an emotional register, but only a certain kind of emotion.
An example of Ti and Fi (I think this is accurate):
Say the task is to choose an item of food
Ti will look at the options, and then weigh up which is the most suitable in a detach way. It determines which option is the "best".
Fi will look at an item of food, and ping it for whether it is "in harmony" with the person, i.e. whether its inclusion will lead to a more harmonious state. Maybe more than one item is felt as desirable, and other functions come in to help determine how to create a balanced item that is inclusive of them all.
And actually, Ti can look at a single item too, with the option being a yes/no. So my explanation kinda falls apart/needs work. Ti is more detached/cool-headed though, whereas Fi is more heart-felt/warm.
I would say perhaps that Fi has a tendency to try to include, and Ti has a tendency to try to exclude.
That's why Ti is good with maths and science. It breaks something down into its parts, while excluding as much information as possible to give a concise model.
Whereas Fi would generally be more useful for creating art, because the Fi determines how to bring disparate parts into the one picture, in accordance with the internal mindstate and beliefs.
There's obviously way more to say on the relation between Fi and emotions but I haven't thought it through.