They looked at what happens 1 month after either infection or vaccination. They continued watching for 6 months.
The vaccinated were about twice as likely to test positive for covid in those 6 months. (This is the part that some may choose to quote. Natural immunity is better!)
Here are three advantages of choosing the vaccine route to protection instead of the catching covid route:
The vaccinated were less likely to have a trip to an emergency department.
The vaccinated were less likely to spend time in a hospital ward.
The vaccinated were less likely to die.
Study:
https://ajph.aphapublications.org/doi/full/10.2105/AJPH.2022.307112
They tried to match anyone who had been vaccinated with someone very similar who had not been vaccinated but who had caught COVID (and survived). They ignored the immediate effects of either infection or vaccination, but concentrated on the next 6 months.