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. :-)

Yes $<$ and $\lneq$ are the same relation.
However, be careful about set inclusion: older texts, and some current mathematicians, use $\subset$ to mean $\subseteq$, and to indicate strict inclusion they might use $\subsetneq$ or $\subsetneqq$. Other authors who use both $\subset$ and $\subseteq$ will use $\subset$ to mean strict inclusion. These days, $\subset$ can be confusing. If you just jump into a book or paper and see $\subset$, it's best to check Section 0 or Chapter 0 to be sure what it means.
So $\subset$ and $\subseteq$ aren't necessarily different relations, universally, though they almost certainly are in a text that uses both. To avoid possible confusion, though at the cost of losing the nice analogy with $<$ and $\le$, it's probably best to stick to $\subseteq$ and $\subsetneq$.
