These two historical periods are quite strongly linked, to the point that a clear division between them doesn't exist. Having studied both, I see the distinction as follows:

Late Antiquity generally refers to the period of the final centuries of the Roman Empire, which was markedly different than the Julio-Claudian or Antonine periods we generally envision, separated from the more iconic early empire by the political and military chaos of the third century.

When I think of Late Antiquity, I see a period in which the Roman Empire was, among other things:

a) increasingly ruled from capitals outside Rome, including Constantinople, Milan and Ravenna, with the almost complete elimination of the power of the senate.

b) increasingly fragmented, including the great East/West split,

c) increasingly Christian, culminating in Constantine's personal conversion and, ultimately and equally importantly, Theodosius' conversion of the empire itself to Christianity and the banning of traditional Roman religious practices,

d) dramatically different in the practices and organization of the Roman bureaucracy, citizenry and army than the empire of the first and second centuries CE,

e) faced with a separate set of foreign policy challenges than the early empire, including the Vandal, Gothic and Hun migrations and the rise of the Sassanid Empire,

f) and often divided by doctrinal debates within Christianity that impacted imperial politics and sometimes ended in bloodshed.

As for the early medieval era, to me it generally suggests the period wherein Western Europe was recovering from the loss of a unified culture and society, fragmented into a series of post-Roman petty kingdoms that had not yet coalesced into the greater kingdoms we all know and love. It's a period marked by, among other things,

a) attempts to preserve the Roman past, limited by lack of infrastructure and and the loss of classical modes of education,

b) a centuries-long debate to define the expanding role of the church in European politics and society, with a far-flung church that was now solidly Trinitarian but not yet the unified institution of the high or late medieval period, and advanced by increasing numbers of influential bishops and monasteries,

c) suffering from increased but not total isolation from other parts of what had been the ancient world,

d) brought to an end in part by the formation of stronger institutions (both religious and secular) and the intellectual advancements of the 'Twelfth Century Renaissance' and the Scholastics who followed.

TL;DR: Late Antiquity is when Romans had to redefine what it meant to be a Roman, the early medieval era is when Europeans had to define what it meant to live in a Europe without the Roman Empire.

IN TWO PARTS FOR REASONS OF LENGTH

The most universal of the "regulative fictions" which historians employ to make some order out of a chaotic past is "periodization," by which we cut the continuous thread of time into manageable lengths, and then do our best to present such divisions as natural rather than contrived. (Peter Novick, That Noble Dream, 16)

Let me repeat for emphasis: all periodizations are fictions. They are anachronistic creations imposed on the past. Novick, as I understand the quote above, would have us believe that they are simply necessary contrivances which historians create, and which they do not really believe. Novick is an eminent historian; his formulation is probably how most, even most historians, tend to assume periodization works.

They are wrong. Periodizations are not our servants but our masters, and the creation and utilization of periodization is necessarily a political act.

In her very difficult but brilliant book Periodization and Sovereigny, Kathleen Davis has recently shown that the idea of a "medieval" age opposed to the "modern"(1) is a product of English juristic disputes over the location and possession of sovereignty in the sixteenth and seventeenth centuries, responding to the problems created by the English Civil War and the decapitation of Charles I. The primary method by which these jurists distinguished between themselves and the past, i.e. between royal and parliamentary authority, was to reify the idea of "feudalism". But what they (and four hundred years of subsequent scholarship) referred to as "feudalism" is derived from a collection of texts called the Libri feudorum (Books of Fiefs), a private, “eclectic collection of treatises, statutes, and northern Italian legislation regarding fiefs... accumulated in various recensions throughout the twelfth and thirteenth centuries.”(2) Thus, the Libri feudorum did not describe a general system, or even a specific one. But the jurists found the Libri to be useful, as it allowed them to separate authoritarian monarchy as an arbitrary, feudal, "medieval" form of government, as opposed to the parliamentary, pluralistic, "modern" one.

The reification of feudalism and the creation of the medieval in the seventeenth century is not simply a distortion of our view of the past; it created a meme that is deeply embedded in our culture. As Timothy Mitchell discusses in Rule of Experts,(3) when the British took control of Egypt in 1882, they determined that "under the laws of the Ottoman Empire, which had ruled Cairo since 1517, all land was considered property of the state."(4) Such a system was (to the report’s authors) not based in an abstract body of law, but in the personal, tyrannical will of an absolute ruler, and thus patently inimical to the conception of private property, which was seen as the cornerstone of (European) civilization. The 'modern' European conception of property directly opposed individual and arbitrary---that is to say 'medieval'---coercive power, of which the state ownership of land was one facet. Consequently, the imposition of "law based on private rights represented a rupture with the previous world of arbitrary and despotic power."(5) The British report advanced the argument that the Ottoman Empire’s failure to evolve systems of private property ownership was a clear sign of the backwardness of the Muslim East and a clear indicator of the benefits offered by colonial rule. As Mitchell puts it, to the colonizers, "Law could claim to be universal, and thus nonarbitrary, only by appearing as the expression of civilization."(6) The expression of modern, western, European civilization, of course—what other sort was there? So the artificial and political creation of the periodization "medieval" and its memetic bond to feudalism was also used for political ends.

But let's come back to how the creation of "medieval" has distorted our view of the past. To list every consequence would far exceed the word limit here, but let's look at an easy one: if the medieval ended with modernity, it must have begun. But where? The historiographic discussion over the origins of "medieval Europe" is generally known as "the debate over the year 1000."

Another question about this periodization. When I hear discussion of how and why Late Antiqity is made separate from early medieval, it is almost always in the context of Western Europe. As /u/Antique_futurist described it, "when Europeans had to define what it meant to live in a Europe without the Roman Empire." Can this periodization be made to fit Eastern Europe or the Middle East? Greece, the Balkans, Anatolia, southern Italy, these are not places without the Roman Empire. Their political history is quite different from that of France, Germany, or England, and yet "medieval" is a word I see used to describe the Byzantines, Ummayads, all the eastern empires around this time as well.