As for "the real difference," I think you have to look at a few examples of the different ways Feeler and Thinkers analyze things in order to get the drift. The following is how I look at these things:
Feeling concerns itself with empathy and Thinking concerns itself with equity. Some topics or issues in the world are going to be best addressed by looking at issues of empathy, in which case Feelers are going to gravitate toward those issues and be capable of doing in-depth analysis there. OTOH, other topics or issues are going to lend themselves to an equity analysis, and Thinkers are going to outperform on such issues.
Example of a Feeler-oriented issue:
Guy X is hosting a party at home, attended by X's best male friend and X's fiancee. The male friend tends to be cantankerous and loud, and at some point during the party the best friend and the fiancee get into a long, heated argument over some issue. X simply stays out of it and lets the two of them duke it out. Later, after the party has ended, the fiancee chews out X for not stepping in and supporting her. X comes to TypoC and asks what he should have done.
Feelers are probably going to have a field day with this one. They'll be trying to harmonize the interests of the three parties, with reference to the obligations of partnership/marriage vs. friendship and pulling in side issues like codependency and whether the fiancee might be trying to isolate X from his friends, and so on. Thinkers, on the other hand, probably won't relate much to the situation and will probably do only a cursory equity analysis: X has substantial ties to both parties; both parties have equal claim to support from X; so the fairest thing is for X to stay out of the dispute, exactly as he did. End of story.
Okay now an example of a Thinker-oriented issue:
Any purely legal issue will do here, for example, capital punishment. Especially if you debate such an issue in it's most purely philosophical/legal form, i.e., as an abstract concept. In that form, most Feelers will probably have an opinion, but it will be fairly cursory: Well, the murderer took the life of someone else, so he has forfeited the right to his own life. If the state really wants to kill him, then why not? End of story. Meanwhile, Thinkers will probably want to do a much more detailed analysis: Prevalence of capital punishment in other countries, methods of capital punishment used by different states, cost of life imprisonment vs. capital punishment, etc.
You get the picture.
Now, you can flip-flop the examples: Turn the story about Guy X into a legal/philosophical debate (friendship obligations vs. marital obligations), and you can get Thinkers to take more of an interest while losing the interest of many Feelers. Then turn the capital punishment debate into a debate about one specific murderer and the specific victims he killed and the feelings and demands of the families of those victims; and at this point the Thinkers will take less of an interest and the Feelers will take more interest.
These are all stereotypes of Thinkers and Feelers, of course. But you get the drift. Basically, Feelers tend to be attracted to the "human interest" angle of any issue because it highlights the facets that interest them the most: Issues involving achieving harmony between specific parties by empathizing with their needs. Meantime, Thinkers tend to be attracted to the abstract, legal/philosophical angle of any issue because it highlights the facets that interest them the most: Issues involving equity between abstract concepts.
You tend to see this when INFPs and INTPs debate. INFPs will often relate some first-hand or second-hand personal experience to make a point; but INTPs will claim that such stories are anecdotal at best and not admissible as argument. Then INTPs will spell out a legal/philosophical principle; but INFPs will claim that such principles are dry and empty without application to some real-life example. And so on.
Anyway, that's how I look at it. Thinking and Feeling aren't necessarily opposites, just as empathy and equity aren't necessarily opposites. Given any specific issue, they can both lead to the same final conclusion (albeit by different routes). OTOH, they are a dichotomy: There seems to be a fairly clear dividing line there in how Thinkers and Feelers deal with issues:
--Feelers are about empathy and tend to prefer specific, real-life issues: All the better to latch onto the "human-interest" angle in the interests of harmony.
--Thinkers are about equity and tend to prefer abstract, legal/philosophical issues: All the better to parse the equitable division of rights and responsibilities of the parties.
Just brainstorming here. YMMV, of course.