Q:

How to see that upper-semicontinuity and supermodularity are equivalent in a supermodular game context? 

One requirement for a supermodular game $(I, \mathbf S, \mathbf u)$ is usually presented in two ways (e.g. in this note):

For $i \in I$, $u_i$ is supermodular in $S_i$, when $s_{-i}$ is fixed, i.e. for $s_i, s_i' \in S_i$
  $$u_i(s_i \vee s_i', s_{-i})+u_i(s_i \wedge s_i', s_{-i}) \geq u_i(s_i , s_{-i})+u_i(s_i', s_{-i})$$

Or

For $i \in I$, $u_i$ is upper-semicontinuous, i.e. for $\alpha \in \mathbb R$
  $$\{s_i \in S_i \mid u_i(s_i, s_{-i}) < \alpha\} \text{ is open.}$$

But they're obviously different.  The former endows $S_i$ with an order structure, while the latter requires the equipment of a topology.  How to understand this difference? 

A:

Your observation about topology and order is somewhat inaccurate.  First definition make use of partial order on $S_i$ while the second make use of topology on it, but both structures are almost always specified by the assumption, that $S_i$ is a sublattise of $R^n$ and so it is equipped with usual subspace topology and partial order.  This is the first point in the definition in the material you have referred to.  So this worries are misplaced. 
However there is some indications (see, for example, chapter 4 in Supermodularity and Complimentarity by Topkis), that the requirement of upper-semicontinuity is stronger and implies the first one.  I'm not ready to provide an accurate proof, however.