Are there different types of paralysis? For example, if my Pokemon got hit with a Stun Spore versus a Thunder Wave. Since they’re completely different types of moves, does it also affect my Pokemon differently other than being paralyzed?

Yes, actually. Although all moves that cause paralysis affect your maneuverability one way or another, how it happens depends entirely on the nature of the move involved. For example, a physical strike (Body Slam, Force Palm, and so on) is less about paralysis and more about either stunning (in the case of Force Palm) or, well, spraining (in the case of Body Slam). While a pokémon affected by these moves can still battle and move about, it just hurts to do so. In their cases, paralyze heal works by easing the pain enough to restore their ability to freely move (in the cases of Body Slam, Bounce, and other full-body moves) or by simply snapping them out of being stunned (in the case of Force Palm).

Then you have non-contact paralysis, such as Glare, Lick, or technically Psycho Shift. In the case of Glare, the target is far too intimidated to move for the most part, and paralysis medications work by essentially lighting a fire within them (slightly literally in the case of cheri berry, given its spiciness), thereby granting the afflicted pokémon enough courage to finish the battle. Lick and Psycho Shift, meanwhile, are supernatural in nature. Psycho Shift causes its victims to think they’re afflicted with paralysis while supernaturally curing the user of its own, while Lick locks a victim into ectoplasm-induced chills. For the former, paralysis medications work the same way as it would with Glare, while for the latter, it works by dispelling a ghost’s supernatural hold on its victim. 

Finally, Stun Spore and all electrical-moves (including Freeze Shock and the electrical component of Tri Attack) are far more dangerous than the above. When it comes to electrical attacks, those target the interactions between a pokémon’s nervous and muscular systems, which in turn cause the status condition’s signature involuntary spasms and seizes. (This is because the interactions between these two systems are founded in electrical pulses: the natural electricity found in everyone’s nervous system is balanced just so, with a specific pattern. An electric shock can change the pattern of these pulses, which can cause a muscle to “think” it’s received a signal from the nervous system to seize when it shouldn’t.) It should be noted that in high doses, this can also affect the heart, triggering arrhythmia or even cardiac arrest, and thus, it’s absolutely imperative to take caution when training electric-types … and, well, teach your electric-types how to minimize the risk of accidentally doing permanent damage to a pokémon.

By contrast, Stun Spore is actually a poisonous spore spore. When inhaled or swallowed, which typically happens when a victim is doused in it, it triggers a reaction that causes a victim’s muscles to seize up, much like in electrical paralysis. (It can also cause a nasty skin rash, but that’s neither here nor there.) In fact, in high enough doses, it can paralyze the respiratory system and essentially prevent you from breathing. On the positive side, a Stun Spore-induced death is actually far easier to prevent than electric attacks, not only because the wind often spreads the attack thin (which also explains why it’s not always a reliable means of paralysis) but also because in order to produce the doses needed to kill someone, a pokémon would have to consciously do it. Pokémon that have Stun Spore in their arsenal can easily be trained to hold back on their spore production, and trainers themselves can take note to not “spam” this move. It’s also somewhat easier to treat, as Stun Spore paralysis responds well to paralyze heal, and it’s just a matter of getting your pokémon to a center and administering a general antivenom. By contrast, an overdose of electrical paralysis may require additional therapy to heal. Still, yes, whereas electric attacks run a risk of killing a pokémon rather quickly (it only takes a high enough shock to send a victim into cardiac arrest), death by Stun Spore is notoriously slow and painful.

Or in short, the answer to your question is … yes. Yes, there are different types of paralysis, and some are both more fascinating and more nightmarish than others.