SARS-CoV-2 : does vaccination provide a better immunity than being sick and recover?

As @MikeyC and @jamesqf pointed in the comments, vaccines are better documented than the actual cases, whereas qualifying the immunity conferred by having recovered from Covid-19 would require large cohort studies.

However, one can follow up thir argument in many ways:

• It would be reallys urprizing, if large-scale cohort studies have not been udnerway or already available by now: while nationwide testing may still be lacking in many countries, masive and repeated testing campaigns in specific regions are not a rarity by now. (see, e.g., here and here)
• Hospitalization is not a necessary requirement for attesting a covid case - one could test unvaccinated persons for the presence of the Covid-specific antibodies.
• There is much still unknown about the vaccines themselves - particularly about whether the immunity conferred is a lasting one, and whether.
• Moreover, the fact that many people might have lived light cases of Covid does not make it easier for attesting the efficacy of the vaccines. Public authorities openly warn that the vaccine might be inefficient and a vaccinated person might be still a carrier of the disease.

In other words, "we don't know" is the most likely answer.

As the OP, I forgo the ethical issues raised by the "green passport" - these are not only those related to the restrictions on the individual freedom, but also the discrimination against the individuals to whom the vaccine is counter-indicated (e.g., allergics, pregnant women, people with weakened immunity, etc.)