Question: How long is immunity from prior case vs. immunity from vaccine?

Answer: We’re still learning.  Some of what we currently know is this:

• Immunity after infection: There is a lot of heterogeneity in how lasting immunity is after infection depending on a host of individual factors, including the severity of the disease.  This means that while one person may be quite well protected for many months, another may not be.  In this vein, just yesterday Lancet published results from a population-based study out of Denmark, Assessment of protection against reinfection with SARS-CoV-2 among 4 million PCR-tested individuals.  The study used Denmark’s comprehensive testing and health information system to assess the degree to which individuals who tested positive in the March-May period were reinfected (e.g. tested positive again) in the September-December period.  Authors found that “protection against repeat infection was 80·5% (95% CI 75·4–84·5),” however protection from repeat infection was much lower in older ages.  This offers compelling evidence that immunity after infection is relatively robust 6-8 months post-infection at a population-level.  Of course, like all studies, this one has limitations.  For example, during the study period new variants of concern had not established themselves in Denmark; we’re still unsure how protected from reinfection an individual is if exposed to a new variant.   
• Immunity from vaccination: Unlike immunity after infection, immunity after vaccination is generally far more robust across the population (e.g. more homogeneity in effective immune response).  Since available data from clinical trials from Pfizer, Moderna, and Johnson & Johnson only span several months post-vaccination, we know that protection in the months after vaccination is robust, but we don’t yet know how long the protection lasts.  We just don’t have the data yet, but as the months and years progress, we will!  When it comes to variants of concern, so far available data indicates that the vaccines offer protection.  If protection were to wane (either in duration or due to a new variant), booster shots would be required.  For more on the variant/vaccine issues, check out the perspective published yesterday in JAMA, COVID-19 Vaccines vs Variants—Determining How Much Immunity Is Enough.