Another big difference is the water volume. When we swim in the ocean, our sweat, urine and fecal matter that we shed is diluted quickly and represents a small fraction of what is in the ocean overall. The ocean is also very salty so not everything that lives inside us does well there, but the same dilution argument holds for fresh-water lakes and rivers as well. Swimming pools are different because we are constantly introducing new bather waste into them and because they are an unbalanced unnatural ecosystem. So without disinfection, uncontrolled bacterial growth can occur. Also, with multiple bathers in a relatively much smaller volume of water, it's more likely for there to be person-to-person transmission of disease (mostly viruses and protozoan oocysts; for bacteria the issue is more one of uncontrolled growth so that we ingest more than our body can handle).
Remember that without disinfection and oxidation of bather waste, we would be swimming in our own sweat, urine (not actually a problem even though it's gross to think about) and fecal matter just like taking a bath using the same water over and over again. Yech!
Read
this document from the CDC if you want to get an idea of various waterborne pathogens with real-life examples. A more complete list is found in the water-related diseases by type on
this page.
The irony is that when it comes to pools, we are the worst contributors to infection. That is, we either directly transmit our disease to someone else or we shed pathogenic bacteria that are "under control" in our own bodies, but grow to much higher concentration in a nutrient-rich pool and then we re-ingest it and can't handle its much higher concentration. If you kept people and other animals out of pools, they'd be a lot safer, but that defeats the whole purpose of having a swimming pool!