What does "sequential memory-hard" mean?
The Scrypt paper here defines memory-hard and sequential memory hard, and accordingly explains why one was used over the other.

Definition 1. A memory-hard algorithm on a Random Access Machine is an algorithm which uses $S(n)$ space and $T(n)$ operations, where $S(n) \in \Omega (T(n)^{1-\epsilon})$


Definition 2. A sequential memory-hard function is a function which  (a) can be computed by a memory-hard algorithm on a Random Access Machine
in $T(n)$ operations; and  (b) cannot be computed on a Parallel Random Access Machine with $S∗(n)$ processors and $S
∗(n)$ space in expected time $T∗(n)$ where $S∗(n)T∗(n) =\mathcal{O}(T(n)^{2-x})$ for any $x &gt; 0$.


Put another way, a sequential memory-hard function is one where not only the
fastest sequential algorithm is memory-hard, but additionally where it is impossible
for a parallel algorithm to asymptotically achieve a significantly lower cost. Since
memory-hard algorithms asymptotically come close to using the most space possible given their running time, and memory is the computationally usable resource
general-purpose computers have which is most expensive to reproduce in hardware
, we believe that, for any given running time on a sequential general-purpose computer, functions which are sequential memory-hard come close to being the most
expensive possible functions to compute in hardware.