By the mid-400s BCE the alliance had become what is called the Athenian Empire, and by 431 BCE the threat it posed to Sparta, with Athens' heavy-handed rule, prompted the Peloponnesian War.
Based on Wikipedia: Delian League
By the mid-fifth century BC, it had developed into a naval imperial power, conventionally called the Athenian Empire, where Athens established complete dominion and the allies became increasingly less autonomous.
By 431 BC, the threat that the League presented to Spartan hegemony, combined with Athens's heavy-handed administration, prompted the outbreak of the Peloponnesian War