Once a cell goes through the process of determination, what happens next as the cells develop in the embryo?

When a cell is differentiated that means it is no longer a ''blank canvas'', a pluripotent cell (like the stem cells).

Let's take a muscle cell as an example. 

Before it is differentiated, it is just like all of the other stem cells.
But after the expression of genes specific for muscle cell and the resulting changes in structure and physiology of the cell, this cell becomes a muscle cell.
Afterward, the differentiated muscle cell communicates integrates with other muscle cells into the muscle tissue. The muscle tissue eventually forms organs like skeletal muscles, the heart and similar.