The only difference, for any display type that has block and inline variants, is that the inline-* display type has the box laid inline (i.e. in an inline formatting context) while the other has the box formatted as a block-level box, subject to most of the same formatting conventions as other block-level elements in a block formatting context. The difference between a block-level box and an inline-level box is covered in depth elsewhere.
Everything concerning how the box lays out its contents is pretty much the same (the specifics of which, of course, are governed by the display type itself); any other nuanced differences would have been stated explicitly in the spec. As far as I'm aware, there are in fact no such differences.
When in doubt, prefer block-level display types. If you find yourself asking whether inline-level is appropriate, chances are the answer is no. Certain scenarios may prevent a box from ever being formatted as an inline-level box anyway, such as absolute positioning or floating, or being formatted as a flex item or grid item instead. The result is a direct conversion from the inline-* variant to its usual block variant. That is, inline-block is converted to block, inline-table to table, inline-flex to flex, and inline-grid to grid. Again, this does not directly affect how an element's contents are formatted, not as far as the specifications go.
Examples of each display type and its inline-level counterpart follow.
In CSS2.1, section 9.2.4 describes block and inline-block as follows:
block
This value causes an element to generate a block box.
inline-block
This value causes an element to generate an inline-level block container. The inside of an inline-block is formatted as a block box, and the element itself is formatted as an atomic inline-level box.
Note that "block box" is a shorthand for "block-level block container", and a block container is something that can contain block-level boxes.
You can see that both of these two values cause an element to generate a block container box, in which its contents will always follow the same set of formatting rules, but that block container box itself is either formatted as block-level, or inline-level.
There is one additional difference between block and inline-block: an inline-block box always establishes a new block formatting context; block boxes only do so under a set of conditions. This does not hold true for any of the other display types that have block-level and inline-level counterparts.
Section 17.2 describes table and inline-table as follows:
table (In HTML: TABLE)
Specifies that an element defines a block-level table: it is a rectangular block that participates in a block formatting context.
inline-table (In HTML: TABLE)
Specifies that an element defines an inline-level table: it is a rectangular block that participates in an inline formatting context).
The Flexbox module describes flex and inline-flex as follows:
flex
This value causes an element to generate a block-level flex container box.
inline-flex
This value causes an element to generate an inline-level flex container box.
And the Grid Layout module describes grid and inline-grid as follows:
grid
This value causes an element to generate a block-level grid container box.
inline-grid
This value causes an element to generate an inline-level grid container box.
Again, in all of these scenarios, a table, a flex container, or a grid container will behave exactly the same way whether it is block-level or inline-level. A flex container always establishes a flex formatting context for its flex items, and a grid container always establishes a grid formatting context for its grid items.
