Religion vs. Science?
The story takes place in the Victorian era, Darwin’s theories had recently been published, science was super controversial. And often, Hyde is referred to as animalistic and such, drawing clear allusions to Darwin’s theory of man. Lanyon was also very against Jekyll’s experiments and often exclaimed to God- he represents the unwavering church.
Another external conflict could be between Jekyll and utterson, who’s really just looking out for him. Utterson tried to help Jekyll see reason but Jekyll pushes him away.
I guess another one could be jekylls struggle with maintaining his status and reputation? Again, in the Victorian era gentlemen we’re super concerned with how they came across to people and stuff, hence the reason Jekyll created mr. Hyde. Hyde represents jekylls struggle to chill out (though I guess you could call that more of an internal struggle?)
I couldn’t remember many but I hope this helps!