First off, there's not actually any evidence that Louis XIV ever used this phrase, and I'm not sure how well it really embodies his brand of absolutism (e.g. what about his famous ministers like Colbert and the unprecedented 50,000 bureaucrats running things even during the early years of his reign?) (this figure from Van Creveld p. 135). But the answer is, yes, the absolutist state was very different from the feudal idea of the state.
My answer will mostly be drawing on Martin van Creveld's 'The Rise and Decline of the State' (1999) (which I highly recommend), and also Philip Bobbit's 'The Shield of Achilles: War, Peace, and the Course of History' (2003), and maybe a few other authors as well.
In a strict sense, there was no feudal idea of the state, as the word 'state' was not in use in the modern sense of a supreme government of a territory. There were empires, kingdoms, duchies etc., and within these there 'estates' (e.g. the estates of the realm), from which the modern concept of the state later derived in the early modern period. Bobbit (ch. 6) dates it to Italy at the end of the 15th century, with this new constitutional outlook described by Machiavelli. Liah Greenfeld, who writes that the modern idea of the state appeared in England earlier than in other countries, dates the modern terminology in English to the the Elizabethan period (Greenfeld, Liah (1996). Nationalism and Modernity. Social Research Vol. 63, No. 1, NATIONALISM REEXAMINED (SPRING 1996) (pp. 3-40) (p. 22). However there's some dispute that the word 'state' was used even in the time of Louis XIV. Greenfeld in 'Nationalism: Five Roads to Modernity' quotes the Marquis d'Argenson writing in 1754 that "never before were the names of Nation and State evoked as often as today. These two words were never pronounced under Louis XIV, and one hardly knew what they meant." (p. 158-159).
I should note at this point that these works are focused on Europe, and when Van Creveld writes that the state came into existence here during the late medieval / early modern period, I would dissent by saying that the Chinese imperial bureaucracy certainly qualifies as a state in his sense.
But looking at Europe, there are two key difference between absolutist states and government in the feudal era. The first difference is that in feudal society, power, both in theory and in reality, was shared between more entities than just the monarchy (and the monarchy, or its supporting institutions, is what what would develop into the state in later years). Van Creveld describes how in the years 1300 - 1648, the monarchs struggled against these competing entities (the church, the (Holy Roman) empire, the nobility, and the towns) to establish their supremacy, and thereby laid the foundations for the state. The second key difference is that the extent of 'what the state could do' expanded during this period (at the expense of these other entities), most importantly regarding governance, taxation and war, and these new capacities are really what we think of as embodying the state today. This has been termed 'state capacity' by some economic historians.
I'll briefly describe some of these changes below, as I said before, I recommend van Creveld's book for more.
In the medieval period, the church had significant political power. E.g. see the investiture controversies of the 11th and 12th centuries, or King John of England's conflict with the papacy, where in both cases the church got the better of (or at least maintained equality with) emperors and kings. Over the next 500 years the power of the church was reduced in favour of the monarch (e.g. see the English Statutes of Praemunire of 1351 and the French Pragmatic Sanction of 1439), with the peace of Westphalia famously ending the hegemonic ambitions of the Catholic Church and the Holy Roman Empire, and replacing this ideal with a system of sovereign states. In newly developing strong states, whether they were protestant like England or Catholic like France, the state eventually subordinated the church.
While the Holy Roman Empire did not have actual power over the kingdoms outside it in Europe, there was an idea that it had nominal supremacy over them, with the emperor at the head of the feudal hierarchy, enjoying precedence when he met with another ruler. The empire declined in relative power from the high middle ages (e.g. the battle of Bouvines in 1214 marked the last serious attempt by a Holy Roman emperor to 'put back the clock to the days of Charlemagne' as van Creveld puts it (p. 75-6)), and as with the church, the peace of Westphalia is seen as the symbolic start of a new era of equal sovereign states.
The late medieval period saw the process of monarchs centralising power and taking it away from the nobility. In England the 'Edwards' started this process, and Henry VII completed it in the late 15th century (e.g. with laws against livery and maintenance and increasing the powers of the Justices of the Peace). In France centralisation was pronounced under Philip IV and his immediate successors, and in the reign of Louis XIV nobles were deprived of their provincial governorships and intendants appointed in their place. The nobles thus came to depend on the state for their power (through government positions), rather than on their own lands and retainers.
As mentioned, in France, the number of government officials had increased from 12,000 in 1505, to 25,000 in 1610, to 50,000 by the early years of Louis XIV's reign. The number of intendants also increased, as mentioned previously, replacing nobles as provincial governors. So now the state actually governed its territory, whereas previously the monarch had only had feudal precedence over self governing areas.
The idea of modern taxation is also a development of this period, as previously, monarchs were expected to live off their own lands, only asking for additional money for wars or other extraordinary expenditures. For example in 1452 Charles VII of France instituted the taille, or land tax, and forbade the nobility from instituting similar levies of their own, thereby making the distinction between rents (for feudal lords) and tax (for the king). In 1523 Francis I published the edicts of St. Blois and St. Germain-en-Laye, establishing a single treasury, and abolishing the distinction between ordinary and extraordinary revenue. In the reign of Louis XIV Mazarin and Colbert worked to increase the ability of the monarch to collect taxes.
So in the feudal period, there wasn't really the concept of the state in the modern sense in Europe at all. Within a kingdom / empire / duchy there were many forms of power, often overlapping and competing, of which the monarchy was just one. In the late medieval and early modern period, the monarchy suppressed the power of these competing forces, and increased its own ability to control its territory through bureaucracy and taxation, thereby bringing into existence what we think of today as the state.