Natural vs. Supernatural Orders
A human is composed of a body and a soul. Since the human soul is virtually infinite—just think of all that man desires, can know, and can do etc.—, it is not fulfilled by finite creation. Man was created for a higher, supernatural end, which he can attain by God's grace.
Your question is essentially asking about the distinction between the natural and supernatural orders—a distinction that is very frequently denied today, especially in the heresy of Modernism (the "synthesis of all heresies"), which Pope St. Pius X condemned in his 1907 encyclical Pascendi Dominici gregis §10:
There is no question now of the old error, by which a sort of right to the supernatural order was claimed for the human nature. We have gone far beyond that: we have reached the point when it is affirmed [by the Modernists] that our most holy religion, in the man Christ as in us, emanated from nature spontaneously and entirely. Than this there is surely nothing more destructive of the whole supernatural order. Wherefore the [First] Vatican Council most justly decreed: "If anyone says that man cannot be raised by God to a knowledge and perfection which surpasses nature, but that he can and should, by his own efforts and by a constant development, attain finally to the possession of all truth and good, let him be anathema" (De Revel., can. 3).
[Those who believe that condemned statement are Pelagians.]
It's true that God is infinitely different and distinct from the created world, but we can certainly know God from the created world (cf. Rom. 1:20). Just as we can know an unseen cause (e.g., magnetism) by studying its effects (e.g., magnetic bodies attracting each together), so we can know God (creation's First Cause) by studying His effects (creation itself).
Think of a stop sign. It is octagonal, red, has some black markings that make sense to those who can see and read. A stop sign is infinitely different than the concept or action of stopping, yet the sign still signifies it.* Analogously, creation signifies or points to God, like smoke signifies a fire,** even though He and creation are infinitely distinct.
*This is just an analogy because creation does not signify God by some conventional sign like a stop sign. **This is also an analogy because smoke and fire aren't infinitely distinct like creation and God are.
But there are some truths, necessary for his salvation, that man cannot know from nature alone—e.g., the Trinity. Hence, supernatural grace is required to raise man to the supernatural order.
The Natural & Supernatural Ends of Man
Man has a twofold end or purpose, which St. Thomas Aquinas describes in Summa Theologica I q. 23 a. 1 c.:
The end towards which created things are directed by God is twofold; one which exceeds all proportion and faculty of created nature; and this end is life eternal, that consists in seeing God which is above the nature of every creature, as shown above (Question , Article  ["Whether any created intellect by its natural powers can see the Divine essence?"]). The other end, however, is proportionate to created nature, to which end created being can attain according to the power of its nature.
Man's earthly life is a prelude to life everlasting.* Thus, there is a continuity between this life and the next, just as there is a unity between the body and soul in a person.
*cf. Life Everlasting by Fr. Réginald Garrigou-Lagrange, O.P. (ebook; video of famous quote from it)