... que está empezando a cansarme.
... que me está empezando a cansar.
Is it just a difference of one being stylistically better than the other? Or do they differ in meaning?
They are perfectly equivalent in modern Spanish. Whenever you have a personal pronoun or verb that's in the infinitive (or in the gerund) and is hierarchically linked to another verb (in this case, empezando, which in turn is linked to está), you are able to "raise" it higher in the chain —that is, bring it closer to the main verb.
You can also partially raise it, and attach it to empezando, although it will require an accent in such a case: "...que está empezándome a cansar".
—Hay algo que está empezando a cansarme.
—Hay algo que me está empezando a cansar.
As @guifa said, there's no difference in meaning.
Interestingly, the other variant can be used when asked for repetition: if you said the former, when you're asked to repeat, you can say the latter.
The choice follows no rules and they are perfectly interchangeable.