So, first and foremost, a basic synopsis of what the Delian League was. It was supposed to be a coalition of city-states led by Athens to strip Persia of its remaining holdouts across the Aegean. Of course, the city-states had to pay (whether in gold or in ships and men) for the pleasure of being a part of the League. As Persian power began to recede and become less of a threat, Athens refused to let city-states leave the league and grew very comfortable with the influx of money and power that the position afforded them. Some city-states realized the disparity of power and protested, threatening to leave, Athens (with the power of the rest of the league behind them) refused to let that happen.

Fast forward to the Peloponessian War, which many would assert (Thucydides, our main source, included) was largely a result of Athenian hegemony and expansionism. We have a few passages that really highlight the difference between the Delian League and "Athenian Empirei"

So the situation is thus, Cleon (an influential Athenian statesmen after the death of Pericles) is arguing that Mytilene should be punished for its rebellion against Athens.

What you do not realize is that your Empire is a tyranny exercised over subjects who do not like it who are always plotting against you; you will not make them obey you by injuring your own interests in order to do them a favour; your leadership depends on superior strength and not on any goodwill of theirs. (Thuc. 3.37)

Note: I was informed by /u/bitparity that the penguin translation I was going off of was innacurate to a degree. Penguin decided to translate the word ἀρχή as empire, though it is more like rule or "living under the power of" (similar to the Roman concept of imperium). All in all, the Athenians never called themselves an empire. However, we must look at their actions and words and determine for ourselves what they were.

The Athenians end up being swayed by his argument to destroy defiant Mytilene, only to change their minds soon after. However, the argument that sways them wasn't an appeal to their better natures, but rather, what Mytilene offered to them if left in tact.

Fast forward again, the slaughter at Melos is particularly revealing to Athenian Imperialism. I also suggest you read the Melian dialogue, its widely available online.

Now, Melos was a tiny island and, although a Spartan colony, had stayed out of the war. Athens decides to destroy Melos (i.e. killing the men, enslaving the women and children, sending new colonists to settle the land) for absolutely no reason. Seriously, you can read the passage and the Athenian delegates propose some pretty bizarre reasoning for the destruction of Melos:

[I]f we were on friendly terms with you, our subjects would regard that as a sign of weakness in us, where your hatred is evidence of our power.” (Thuc. 5.95)

And the ever infamous:

“[T]he standard of justice depends on the equality of power to compel and that in fact the strong do what they have the power to do and the weak accept what they have to accept.” (Thuc. 5.89)

You are welcome to interpret the passages as you please and always bear in mind that Thucydides, our source, was an exiled Athenian general for his defeat at Amphipolis. It is also important to consider the vicious debate between the Athenian statesmen concerning Mytilene, who had rebelled, and the complete lack of debate concerning Melos who had done nothing to their aggressors. I would argue that the longer Athens sat on top, the more comfortable they became with their imperial aspirations and inclinations. The line between Athenian headed league and Athenian Empire was thin and as the realities of war set in, the line became thinner and thinner until it ceased to exist at all. They may have begun as a city-state that dominated a league of quasi-equals, but near the end, the interactions were more akin to what you would see between a master and a slave.

As I've mentioned with the Melian dialogue, the Mytilenean debate is also readily available online. Feel free to read them for yourself and observe the difference between the two when they were only just a decade or so apart.