There reaches a point where leadership requires you to take the manager gloves off and get your hands dirty. For small teams or small problems, a good leader/manager can be both one and the same.
However as the size of the team or problems grows, it becomes more difficult (think n^2) to tackle both issues at the same time with the same person.
Management issues (firing, hiring, expenses, budgets) and Leadership issues (decisions, progress, advocacy, examples) on a team of 5 may be doable by one person, but may be stretched with one person servicing a team of 15 or 150.
Or if there is a ton more problems in one area vs. others, the challenge of dealing with that problematic area can leave a gap on the "less problematic" areas, which eventually turn them into actual problems. That's my guess as to what people complain about the most.
Management is inward facing, paying attention to the people you are responsible for, and knowing the right thing that's needed at the right time.
"If you want to build a ship, don't drum up people to collect wood and don't assign them tasks and work, but rather teach them to long for the endless immensity of the sea." - Antoine de Saint-Exupery
Leadership is building the boat, management is making sure the boat gets built, on time, on budget, with the proper permits and inspections, training and safety gear, along with bi-weekly status reports to customers.
Some people confuse one with the other, but to me they're distinct and this is a rough sketch of how I see them as different.