I have found no satisfactory answer anywhere online or in any manuals. What is the difference between a simple weapon and a martial weapon, really?
The difference is in how easy they are to use.
The "Weapon Proficiency" section of the rules says:
The two categories are simple and martial. Most people can use simple weapons with proficiency. These weapons include clubs, maces, and other weapons often found in the hands of commoners. Martial weapons, including swords, axes, and polearms, require more specialized training to use effectively. Most warriors use martial weapons because these weapons put their fighting style and training to best use.
Which implies that martial weapons may be more powerful or better able to deliver on a trained warrior's skills, but that simple weapons are more commonly found and easier to use.
The difference is in their primary purpose
Simple melee weapons are typically implements used for primarily crafting/agricultural work. Sickles are for harvesting. Javelins for picking up trash or putting up tents. Quarterstaffs for walking and slinging multiple baskets of fruit/water. Handaxes are for splitting wood, etc, etc, etc.
Martial weapons are designed to be used primarily as weapons. Sure you can bale hay with a Trident, that doesn't mean that was it's primary purpose. Also I wouldn't recommend it. The blades on a Trident would shred a bale of hay.
Under the categories of weapons, the reference to commoners in the "Weapon Proficiency" section is pertinent to this distinction:
Most people can use simple weapons with proficiency. These weapons include clubs, maces, and other weapons often found in the hands of commoners.
Commoners are typically your field hands, blacksmiths, cooks, cobblers, stableboys, etc. The reason they'd be proficient with these "weapons" is because they use them day to day in their jobs. Whereas Martial Weapons aren't something commoners would typically just have lying around. They'd be made for specialized jobs such as guards, hunters, knights, scouts, etc.
With the intended purpose of inflicting harm, special training is also required for the use of Martial Weapons, which is also why in numerous places in the manual, Martial Weapons proficiency is granted by classes typically associated with those jobs.