What is the difference between a semiconnected graph and a weakly connected graph?

$$bulletlongleftarrowbulletlongrightarrowbullet$$

From those definitions, the above diagram is weakly connected, but not semi connected, as you cannot get from either end node to the other, following the directions of arrows.

Semiconnected means that for every pair of vertices $(x,y)$, either there exists a path from $x$ to $y$ or a path from $y$ to $x$, with all steps of the path obeying the directionality of edges.

Weakly connected means that if you replace all the directed edges with undirected edges then the resulting undirected graph is connected.

Semiconnected implies weakly connected. The converse is not true, for example the graph $A rightarrow B leftarrow C$ is weakly connected but not semiconnected, because there is no path between $A$ and $C$ in either direction obeying the directionality of the edges. But if you forget about the directionality entirely then the graph becomes just $A leftrightarrow B leftrightarrow C$ which is definitely connected.