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The work of Breeden and Litzenberger-formula (https://papers.ssrn.com/sol3/papers.cfm?abstract_id=2642349) gives us a risk neutral probability distribution of a stock price, depending on the option prices.
Question 1: Is there a theoretical or practical way of obtaining real world probabilities given risk neutral ones?
Question 2: What does a real world probability distribution look like compared to a risk neutral, in empirical cases?
Great question! Unfortunately, it's not easy. We can use option prices to get the $\mathbb{Q}$-distribution. However, the probability measure $\mathbb{Q}$ merges the stochastic discount factor (SDF) $M$ and the real world probabilities, $\mathbb{P}$, and it's not clear how to untangle the two (see this answer). Essentially, you have one equation, but two unknowns.
You can recover $\mathbb{P}$ if you make assumptions about the SDF $M$, see this answer for an example (power utility). However, the asset pricing literature has many, many different models for the SDF and a misspecified $M$ gives you misspecified real world probabilities...
There is hope though. ``Recovery theory'' is part of current research. Famously, Steve Ross proposed a possibility in his 2015 JF publication. However, Jackwerth and Menner (2020, JFE) cast doubt whether the recovery theorem is compatible with future realised returns and variances. So, this recovery is still being researched.
How do $\mathbb{Q}$ and $\mathbb{P}$ differ? Well, the answer is the SDF. You can take the simplest models (log-normal) to get a taste, see this answer. Essentially, the likelihood of bad events get inflated under $\mathbb{Q}$ because these are the states with high marginal utility (= that investors fear) whereas $\mathbb{Q}$ puts less weight on good events.
