I think the distinction is made more for conceptual reasons, which has practical implications, so let me review the usual definitions of a stochastic and partially observable environment.
A stochastic environment can be modeled as a Markov Decision Process (MDP) or Partially Observable MDP (POMDP). So, an environment can be
- stochastic and partially observable
- stochastic and fully observable
The stochasticity refers to the dynamics of the environment and, more specifically, to how the environment stochastically moves from one state to the other after an action is taken (basically, a Markov chain with actions and rewards). In other words, in a stochastic environment, we have the distribution $p(s' \mid s, a)$ (or, in some cases, the reward is also included $p(s', r \mid s, a)$). If $p(s' \mid s, a)$ gave a probability of $1$ to one of the states and $0$ to all other states, we would have a deterministic environment.
The partial observability refers to the fact that we don't know in which state the agent is, so we can think of having or maintaining a probability distribution over states, like $b(s)$. So, in the case of POMDP, we not only are uncertain about what the next state $s'$ might be after we have taken $a$ in our current state $s$, but we are not even sure about what $s$ currently is.
So, the difference is made so that we can deal with uncertainties about different parts of the environment (dynamics and actual knowledge of the state). Think about a blind guy that doesn't have the full picture (I hope this doesn't offend anyone) and think about a guy that sees well. The guy that sees well still isn't sure about tomorrow (maybe this is not a good example as you can argue that this is also due to the fact that the guy that sees well doesn't know the full state, but I hope this gives you the intuition).
Of course, this has practical implications. For example, it seems that you cannot directly apply the solutions that you use for MDPs to POMDPs. More precisely, for an MDP, if you learn the policy $\pi(a \mid s)$, i.e. a probability distribution over actions given states, if you don't know the state you are in, this policy is quite useless.
To deal with the uncertainty about the state the agent is in, in POMDPs, we also have the concept of an observation, which is the information that the agent gathers from the environment about the current state (e.g., in the example of a blind guy, the observations would be the sounds, touch, etc.), in order to update its belief about the current state. In practice, some people tried to apply the usual RL algorithms for MDPs to POMDPs (see e.g. DQN or this), but they made a few approximations, which turned out to be useful and successful.
If the difference wasn't still clear, just take a look at the equation that can be used to relate the belief state and the transition model (dynamics) of the environment
$$
\underbrace{b^{\prime}\left(s^{\prime}\right)}_{\text{Next belief state}}=\alpha \underbrace{P\left(o \mid s^{\prime}\right)}_{\text{Probability of observation }o \text{ given }s'} \sum \underbrace{P\left(s^{\prime} \mid s, a\right)}_{\text{Transition}\\ \text{model}} \underbrace{b(s)}_{\text{Previous belief state}}
$$
So, in a POMDP, the policy, as stated above, in theory, cannot depend on $s$, but needs to depend on $b(s)$, the belief state, i.e. a probability distribution over states.
If this answer wasn't still satisfactory, although you probably already did it, you should read the section 2.3.2 Properties of task environments of the AIMA book (3rd edition). Their description of stochastic and partially observable environment seems to be consistent with what I wrote here, but maybe their description of a stochastic environment is not fully clear, because they say
If the next state of the environment is completely determined by the current state and the action executed by the agent, then we say the environment is deterministic; otherwise, it is stochastic
The unclear part is completely determined. They should have said deterministically determined (which you can use for a rap song).
However, they later clarify their definition by saying
our use of the word "stochastic" generally implies that uncertainty about outcomes is quantified in terms of probabilities
In addition to that, they call an environment that is either stochastic or partially observable uncertain. It makes sense to do this because uncertainty makes the problems harder, so we can differentiate between certain and uncertain environments.
To be honest, I don't know if there's some kind of mathematical formalism that doesn't differentiate between stochastic or partially observable environments, but I am not sure how useful it might be.
