Lets use an analogy. Let say a computer program is a Drill. It is a tool used for a specific purpose. It drills hole and screws screws. The computer programmer knows how to assemble a drill. He knows he needs a motor a power supply and a trigger to turn it on and off. The programmer knows how to use the drill and he probably even knows a why the motor spins when electricity is fed through it. The computer scientist designs new a better drills. Making new drills cut through material easier, use less power ect...
There is a large overlap between computer scientist and programmers. Typically computer scientists are focused on areas of research and development (faster search engines, better compression algorithms, better encryption methods, other math heavy stuff) while programmers are focused on practical software development (websites, databases, commercial software. apps).
Even with this overlap I know lots of computer scientists who can't program and I know lots of computer programmers who can't maths.
If you really want to pursue a higher level of understanding I would recommend taking any Statistics, Graph Theory, Number Theory, Dynamic Programming, Analysis or Group Theory course you can find. After all what is Computer Science but applied math.