Good question! I can't talk about ancient empires but rather about how modern colonialism can be seen as different from earlier forms. The empire is certainly the most important state form in history – dominant even until the mid-20th c. according to Frederic Cooper. And many such empires had colonies. Scholars have identified various differences between these older and the modern forms of using colonies: among them the European (and later US and Japanese) belief in one's superiority, due to religious and later racial justifications.

This belief led to attempts to replace local societies and cultures with European counterparts – attempts that proved only partly successful luckily, with indigenous cultures continuing to thrive. Still this marks a major difference with earlier and other, non-European empires. Below I'll draw especially on Jürgen Osterhammel's work on this, which I've found useful (based on earlier answers of mine here). It's by no means the only model thougjh and meant more as a start for others to add to.

For Osterhammel a few factors that set modern European colonialism apart include: at least in theory, the aim to control other states/groups from far away, and to focus solely on the metropole's interests. The colonizers were also generally unwilling to accomodate the native cultures - s.th. not very common in human history. Rather the latter were excepted to "acculturate" to European values. What we do find in most earlier empires instead is that local elites are left in place when they accept the new conquerors' rule, with imperial direct control becoming thinner quickly away from the imperial centre (as eg. C.A. Bayly argues).

And lastly, again, these were generally tied to an ideology of superiority. This included divisions between supposedly "civilized" Europeans and the other "barbarians" or "savages". In connection, modern colonialism went hand in hand with and was made possible through modern slavery of people seen as inferior, esp. Africans but also various native American and Asian peoples. So I think there are many reasons to set European colonialism apart from earlier conquests and colonies, both in how it was carried out and in European self-understanding.

Right, let's look at this in more detail!

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Jürgen Osterhammel argues that a non-Eurocentric colonial historiography should focus not so much on the colonial politics of the European powers, but rather on the rise and fall of specific colonial societies, or social forms - with a stronger focus on the subjects, and interactions between all participants (colonisers and colonized). Because of this there can not be one "history of colonialism", but different histories of forms of colonialims. This is one attempt to go against earlier, more national ways of writing history.

A definition

Before we're taking a look at Osterhammel's ideas for classifiying forms of colonies, his definition of colony gives a good background. Note that I'm drawing on his "Kolonialismus: Geschichte, Formen, Folgen" which has been translated into English:

A colony is a newly formed political construct, created through invasion (conquest and/or settler colonisation) and building on pre-colonial conditions. Its foreign dignitaries are in continuous dependancy relations to a far away "motherland" or imperial centres, which raises exclusive claims of ownership to the colony. [my transl.]

We should note that "colony", „colonialism“ and "colonisation" are not one and the same - there can be colonisation without the creation of colonies; and vice versa, colonies can be created merely through military conquest. Colonisation is this process of aquiring colonies; colonialism is a whole larger system built on the exploitation of colonies, more of which below.

For your question, this would mean that ancient empires certain held colonies, including the Greek who coined the term. But that their form of colonisation was quite different in some ways from modern colonisation – see here especially the „far away motherland“ and its exclusive ownsership claims, and the view of one' superiority I mention early on.

A classification

Osterhammel proposes a classification of colonies that were formed in modern times through the expansion of the European states, the USA and Japan. As noted, these do not mean necessarily a distinction between specific colonial empires, but rather between forms of colonies - some of them shared by the same empire.

1) Dominions ("Beherrschungskolonien")

These were mostly the result of military conquest, often after prolonged phases of contact. They served economic extraction (e.g. through monopolies, tributes), and the strategic protection of imperial policies. Dominions would have a relative small colonial presence, esp. through civil bureaucrats who would return "home" after their service, through soldiers and salesmen - but not solely through soldiers. The governemt was led autocratically through the metropole - through the governor system -, with elements of paternalist care for the native populations.

Examples would include British India, Indochina (Fr.), Egypt (Br.), Togo (Ger.), Philippines (Am.), and Taiwan (Jap.). A variant of this was Spanish America, were European migration led to a mixed, urban society dominated by a creole minority - I'll talk about this some more in the 2nd part below.

2) Foothold colonies ("Stützpunktkolonien")

These were formed as a result of naval actions. They served the indirect commerical exploitation of a hinterland, or as an aid for the logistics of developing maritime power over formally independent states ("gunboat diplomacy").

Examples include Molucco (Port.), Batavia (Dutch), Hongkong, Singapore, Aden (al Br.), Shanghai (international). I'd add here that at least with Dutch colonisation it's often mentioned that it was more focused on exploiting existing trade networks rather than on evangelisation - a marked difference to e.g. Iberian colonisation in the same regions.

3) Settler colonies ("Siedlungskolonien")

These were formed as a result of colonial processes aided by the military . They were supposed to lead the use of cheap lands and cheap (foreign) labout; and make possible the socio-cultural practices of minorieties that were questioned and/or persecuted in the motherland. Colonial presence was primarily through permanent settlers and farmers. Early on this would include the beginnings of self-governance of "white" colonists while violating the rights and interest of the indigenous populations.

Here Osterhammel lists 3 variants:

a) The "New English" type: Displacement and partial desctruction of the native populace - as in the New English colonies, Canada (fr./br.), and Australia.

b) "African" type: Economic dependancy from native labour - e.g. in French Algeria, South Rhodesia (Br.) and South Africa.

c) "Carribbean" type: The import of foreign slaves for labout - in the French & British Carribbean, as well as Brazil (Port.).

While there are major differences between how these modern colonial empires worked, they did have the dependency of the colony to the metropolis in common. This is a relation of ownership and dependency - usually built on the belief in a European/Western "superiority" -, one that works differently and is more pronounced than in earlier colonial situations. As the examples show, this was closely tied to and made possible violently through modern slave labor, esp. chattel slavery; and the redirection of local economies to service the (Western) metropoles.

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To conclude, I would add that to me Osterhammel's classification is useful in that it shows larger trends of colonialism that were common to different regions - often regardless of which European nation/empire held control over the specific region (it's also more complex than what I've discussed here, so I'd recommend the book to those interested). Then again, I find it important to look more closely at the regions themselves to distinguish more clearly between differences and parallels between diffierent colonial approaches.

Last but not least, it's extremely important not to see colonial expansion solely from a European perspective. For many indigenous people colonisation was one among many influences on their lives - a distruptive one for sure, but often not the dominating one. What is more, throughout human history there were many non-European empires that would often clash or collaborate in different ways with the European colonial powers. In many ways, especially "informal" colonialism continues to this day through less obvious, neo-colonial approaches of exerting control.

 

Obligatory NB: I'm proposing one model of looking at colonialism, and don't pretend to be an expert on all the colonial situations/regions it describes; nor for that matter on ancient history (as I mention above).

edit: a word

The main distinction between colonization in antiquity and early-modern colonization and late colonialism (1875-1939 roughly) is more their intellectual consistency than some innate difference. As you reference, all three to involve migration to some degree or another, but we should focus on the motivations for those migrations for why historians view them as distinct.

First, because I am not a classicist I will not feign deep knowledge in Rome, Chinese, Phoenician, or Persian “colonies” but I would count them distinct from the specific word you used, ‘colonialist’ which generally refers to 19th century Empires. Why? Because in the recent years the study of late Imperialism have revealed the importance that colonialist discourse played, while focusing less on the ‘annexing Africa and Asia for resources to feed European industrialism’ approach. I think the majority of your question can be answered by teasing out just what that discourse consisted of.

So what is colonialism? The word colony, colonize, etc. have been employed to varying degrees to describe varying migrations of people from one place with more or less a political attachment to the polity ruling over the ‘mother country’. This is not what most historians would call colonialism, which is a wide swathe of ideas used to justify, encourage, and carry-out the construction of European empires elsewhere in the globe. A good example in the early days of European colonization of the Americas, it was not uncommon to justify the seizing of native land under religious pretenses, imagining that things like forced baptism were helping these people to heaven as opposed to trying to exploit. These notions painted the conquest, plunder, capture and exploitation of the native peoples of the Americas as righteous and good.

To be more specific (and pivot to a subject I know much more intimately) let us consider Late Imperialism in the later 19th century. Colonialism (or imperialism - the terms are roughly synonymous) motivated for a rapid conquest of most of Africa and Asia. Often this expansion was justified in two ways: economic imperative and the so-called ‘civilizing mission’. The economic justification was more than just a need for raw materials. The looming problem of industrial overproduction motivated many industrial nations to imperialism so they would have new markets to sell to, as was one of the large focuses of the US annexation of the Philippines. Also increased competition and protectionism in Europe meant that you didn’t want a colony falling into the hands of a rival, which is why Britain supported Portugal and agreed to King Leopoldo of Belgium getting the Congo - less territory that Britain has to pay to maintain but won’t be taken by France or Germany.

The discourse of the civilizing mission was equally varied. There were some missionaries who genuinely meant well involved with the project of European (and American) imperialism, even if we now acknowledge that they were trying to destroy the cultural and religious heritage of colonized areas. That said, there are far nastier sides of that discourse. As Adam Hochschild brings out in the publicly popular book King Leopold’s Ghost many of the programs in the Congo intended to encourage the “work ethic” of the native population were, in truth, brutal methods of coerced labor to encourage rubber extraction. These justifications were predicated on pseudosciencentific claims that white Europeans were racially superior to African and Asian peoples. This superiority was ‘evidenced’ by the technological and philosophical progress of Europe by the 19th century. How this sense of superiority played out depended on the colony. In his book, The Devil’s Handwriting, George Steinmetz shows that German colonialists approached the hegemony of Samoa as an untouched, natural paradise (including the native Samoans) to be preserved by Europe while at the same executing a staggering amount of Herero after a rebellion in Southwest Africa, to such an extent that many historians call it genocidal. Paternalism or brutality, at the core of European colonialism was an assumption of European supremacy.

This really brings me to the crux of your question, historians rarely connect older migration to the word colonialism because it really does not have the same dynamics of power that later imperialism would have. The colonies of the Phoenicians were not motivated by the assumed inferiority of the people living around modern day Tunisia. While it’s hard to speak generally, the legacy of brutality and violence that were so closely associated with Late Imperialism don’t really resemble the phenomenon of migrations of antiquity.