Q:

When are typefaces considered conflicting with each other?

I have been reading a lot about font pairing lately, but I’m still missing to identify when fonts don’t complement and contrast with each other, instead they clash.
An example that I've read was about comparing a so-called blackletter with a roman serif. They had different proportions, one condensed with large x-height, the other more extended and taller. As they shared the same weight, size and decoration, they were considered too alike. Why was that so, if they had about the same contrast and similarities? 
Could the overall thickness have something to do about it?

A:

Subtle differences look like careless mistakes and sloppiness, not just in fonts but in all designs.
When things are just slightly off, its enough people notice, but not enough people think its a deliberate thought out decision.
Here is Arial Black with Source Sans Pro Semibold. They clash because they're both trying to be clean sans-serif fonts and in that sense have a lot in common. But then there's jarring differences when used together. Just look at those lowercase e's at the end and the angles don't match at all. the o's likewise are fairly different and that's really because they are very different.

Here they are in Regular weight and same size / kerning:

Take either one on its own and its fine. But when used together those little differences make it jarring to look at. Here we can demonstrate this by using both in a sentence. They are similar enough that its hard to tell which words are in which font, but different enough to see that something is off:

If we had instead used a more Contrasting font however it could work even within a sentence:

In some respects it is a matter of opinion, there are many fonts that can pair. But the reasons not to pair is a bit easier to pick out.

Can you easily tell the two fonts are different?

That's really all there is to it. And its important to note this is true of all areas of design. Its why in fashion when someone wears two close but not identical shades of a color its said to clash, why suits are made from the same fabric so the exact same shade is used on the jacket and pants. Or in spacing why it would be very strange if you had 3" then 2.8" then 3" then 3" then 3.1" or something like that.

As far as the exact fonts in your article are concerned. It does a pretty good job explaining it.

While not the worst pairing ever it does clash because they're similar enough that its not immediately apparent that they're different fonts. Especially (and this was probably deliberate in their example) when there is no lowercase a which is the greatest difference between them.

A:

DA01 and Dom are pointing at the central problem:  
They conflict when they don't look good together.
So how do you know what looks good? That's such an obnoxiously subjective thing. But, like so many areas of design, typography is subjective. You have to train your eye for this soft skill the hard way.
Go be a typographer

Set a lot of type
Let pros criticize your work
Fail miserably
Repeat

Eventually (hopefully) you'll just know. And at some point, you'll be able to see the harmonies before you set a single line just by analyzing the structure of the faces.
Study the masters
You'll do yourself a great service to study the masters and the history of type design and typography. 
Two books I consider required for any designer working with type:

The Elements of Typographic Style, by Bringhurst. This one is a virtual encyclopedia for typographers. Bringhurst has a lot to say about proper pairings.
The Form of the Book, by Tschichold. Unfortunately, this one has climbed dramatically in value. But this quick read will teach you some of the most minute intricacies of setting fine type. It's not so much about pairing but, understanding typographic detail and appreciation.

Getting started
So this all begs the question, what qualities of type do you need to understand to build that typographic instinct? In short, everything. But it's a progressive subject. 
Start with fundamentals 

Stylistic categories. This happens to be anything but a settled subject, but there are generally accepted groups like Venetian, Garalde, Transitional, Modern, Scotch Roman, Grotesk, Geometric, Humanist, and so on.
Anatomy. Obviously, the structure of a face is the driver behind all classification. But beyond the hard lines of historical groupings, there are endless ways to combine of these features. The sum of the parts of two faces will dictate the personality and the potential harmony.

As you begin to understand how typefaces are built, you'll start to see why one pairing works and another doesn't. You'll be able to look at a face and say, "it's generally round proportions and angled terminals combined with a slightly tilted axis would make it a perfect partner for ...". That's when things get fun.
Context is everything
If you're setting two fonts at totally different scales with plenty of space between them, you'll be primarily concerned with "personality". The interaction of the details is less critical.
If you're looking for a section header face to pair within text or one face that will be used for in-line emphasis with another, the details become critical. At that point, you probably want to normalize their x-height (size them so their lower case characters line up), make sure their extenders are relatively close, and think about things like crossbar alignment and terminal positions. That's much harder to do well.
To illustrate the point, consider the different requirements between these two scenarios. In both cases, the pairing is Bell and Akzidenz.

Bottom line
Like any craft, typography and typographic pairing are skills built over time and experience. The good news is, just pursuing this in the first place means you're ahead of 80% of your peers.

A:

Make sure they aren't too similar.
Make sure they aren't too different.
Break both rules when necessary (at your discretion).

