Use of the symbol $\lneq$ This answer uses kind of a "proper less equal" symbol: $\lneq$
I would have expected that $<$ is sufficient, because it seems to be the same relation.
For $\subset$ vs $\subsetneq$ some authors interpret $\subset$ allowing self-inclusion $A \subset A$ to be true, while others do not, so the extra symbol offers less ambiguity for more ink.
What is this typically used for?
Update: We have $\ngeq$ too. :-)

Just a speculation. In case of partial ordering (for example, with respect to a cone) there are normally more situations than just $\ge$ and $>$. It may happen that a vector $a\ge 0$ (in the cone), $a\ne 0$ (not a vertex), but not yet $a>0$ (in the sense of the cone interior). It is where the demand for more notation symbols is most likely to come from.
