There is a huge difference.
Computer science is practically a branch of mathematics, or at least was classically seen to be. It encompasses:
- Discrete math (graph theory, combinatorics, logic)
- Programming language theory (type theory, compilers, languages)
- Information theory (crypto, errors, compression)
- Theory of computation (complexity, crypto, automatons)
- Algorithms & data structures (can also include complexity via algo analysis)
- AI (machine learning, robotics, pattern recognition, data mining)
- Parallel & distributed computing (concurrency, communication/networks, HPC)
- Architecture and systems (digital logic, OS, networking, systems arch.)
- Computational fields (scientific computing, numerics, could also include