According to Dayan Grunfeld's lengthy introduction to Horeb (pp. cxx-cxxix), R' Hirsch's approach to mitzvos starts from notes in the margins of his Zohar. (I spoke to an eye-witness of those notes, who saw them when RSRH's Zohar was auctioned off in 2008.)
The difference is that to Rav Hirsch, the Zohar speaks in metaphor, not of metaphysical ontologies.
The Nefesh haChaim, to pick one example, (sec. 1) takes mitzvos as realigning the physical forces of the world with loftier metaphysical ones, (sec. 2) prayer as drawing down holiness to the world, and (sec. 3) learning as a means of bringing Hashem's presence to the studier, and thereby into the world. to him, the acts actually move around qabbalistic "things". To R' Hirsch, all of this is to change one's attitude toward the world by manipulating representations, symbols, of loftier ideas.
The Ramchal, as he explains in the beginning of Qela"ch Pichei Chokhmah, also saw the Zohar and the Ari's Qabbalah as speaking in a symbol system.
I left the Rambam out of that comparison because I believe that his system of explaining mitzvos does not speak to us anyway. It contradicts Qabbalah, Chassidus, Mussar, Rav Hirsch, etc... Now to justify that seemingly bold claim:
According to the Rambam, redemption lays in theological knowledge; not knowledge of G-d, which he believes is impossible, but knowledge about G-d, the philosopher's sort. (See last chapter of Moreh Nevuchim; his words are translated: "דעות אמיתיות באלוהיות" -ibn Tibon, "השקפות אמיתיות בעניני האלוהיים" -el Qafih ["Kapach"], "דעות אמיתיות במטפיזיקה" -Schwartz, "true metaphysical opinions as regards God" -Fraedlander)
It is this knowledge which gives prophets their prophecy (2:12) and earns one individualized Divine Providence (3:18) and to be truly called a person (idid & 3:54) This return from emotion and moral calling to intellectual clarity is undoing the effects of the Tree of Knowledge of Good and Evil (1:1-2). Moral decision-making is less significant than determining Truth. (ibid, 3:54 -- ie the Moreh opens and closes with it.)
Therefore, the Rambam sees all of the mitzvos as serving one of three goals (3:27, tr. Fraedlander):
THE general object of the Law is twofold: the well-being of the soul, and the well-being of the body. The well-being of the soul is promoted by correct opinions communicated to the people according to their capacity. Some of these opinions are therefore imparted in a plain form, others allegorically: because certain opinions are in their plain form too strong for the capacity of the common people. The well-being of the body is established by a proper management of the relations in which we live one to another. This we can attain in two ways: first by removing all violence from our midst: that is to say, that we do not do every one as he pleases, desires, and is able to do; but every one of us does that which contributes towards the common welfare. Secondly, by teaching every one of us such good morals as must produce a good social state. Of these two objects, the one, the well-being of the soul, or the communication of correct opinions, comes undoubtedly first in rank, but the other, the well-being of the body, the government of the state, and the establishment of the best possible relations among men, is anterior in nature and time. The latter object is required first; it is also treated [in the Law] most carefully and most minutely, because the well-being of the soul can only be obtained after that of the body has been secured.
... The second perfection of man consists in his becoming an actually intelligent being; i.e., he knows about the things in existence all that a person perfectly developed is capable of knowing. This second perfection certainly does not include any action or good conduct, but only knowledge, which is arrived at by speculation, or established by research.
It is clear that the second and superior kind of perfection can only be attained when the first perfection has been acquired; for a person that is suffering from great hunger, thirst, heat, or cold, cannot grasp an idea even if communicated by others, much less can he arrive at it by his own reasoning. But when a person is in possession of the first perfection, then he may possibly acquire the second perfection, which is undoubtedly of a superior kind, and is alone the source of eternal life.
Also I would note that the Rambam says that ta'amei hamitzvos can only be explored about the big picture. He writes (3:26):
All of us, the common people as well as the scholars, believe that there is a reason for every precept, although there are commandments the reason of which is unknown to us, and in which the ways of God's wisdom are incomprehensible. ... The repeated assertion of our Sages that there are reasons for all commandments, and the tradition that Solomon knew them, refer to the general purpose of the commandments, and not to the object of every detail.
Those who trouble themselves to find a cause for any of these detailed rules, are in my eyes void of sense: they do not remove any difficulties, but rather increase them. ... You ask why must a lamb be sacrificed and not a ram? but the same question would be asked, why a ram had been commanded instead of a lamb, so long as one particular kind is required. The same is to be said as to the question why were seven lambs sacrificed and not eight; the same question might have been asked if there were eight, ten, or twenty lambs, so long as some definite number of lambs were sacrificed.
This focus on intellect, reducing character (middos, ethical behavior, etc...) to its handmaiden was the topic of major condemnation by many in consequent generations. Rav Hirsch cites the Rambam's inability to apply his philosophy to such details as a symptom of the Rambam's whole system being off. To quote the 19 Letters of Ben Uzziel (Letter 18):
The age gave birth to a man [R’ Drachman’s footnote: Maimonides], a mind, who, the product of uncomprehended Judaism and Arabic science, was obliged to reconcile the strife which raged in his own breast in his own manner, and who, by proclaiming it to the world, became the guide of all in whom the same conflict existed.
This great man to whom, and to whom alone, we owe the preservation of practical Judaism to our time, is responsible because he sought to reconcile Judaism with the difficulties which confronted it from without instead of developing it creatively from within, for all the good and the evil which bless and afflict the heritage of the father. His peculiar mental tendency was Arabic-Greek, and his conception of the purpose of life the same. He entered into Judaism from without, bringing with him opinions of whose truth he had convinced himself from extraneous sources and he reconciled. For him, too, self-perfecting through the knowledge of truth was the highest aim, the practical he deemed subordinate. For him knowledge of God was the end, not the means; hence he devoted his intellectual powers to speculations upon the essence of Deity, and sought to bind Judaism to the results of his speculative investigations as to postulates of science or faith. The Mizvoth became for him merely ladders, necessary only to conduct to knowledge or to protect against error, this latter often only the temporary and limited error of polytheism. Mishpatim became only rules of prudence, Mitzvoth as well; Chukkim rules of health, teaching right feeling, defending against the transitory errors of the time; Edoth ordinances, designed to promote philosophical concepts; all this having no foundation in the eternal essence of things, not resulting from their eternal demand on me, or from my eternal purpose and task, no eternal symbolizing of an unchangeable idea, and not inclusive enough to form a basis for the totality of the commandments.
He, the great systematic orderer of the practical results of the Talmud, gives expression in the last part of his philosophic work to opinions concerning tlie meaning and purpose of the commandments which, taking the very practical results codified by himself as the contents of the commandments, are utterly untenable cast no real light upon them and cannot go hand in hand with them in practice, in life, and in science…
What then is RSRH’s complaint? That the Rambam was too Aristotelian, and it led him to study Judaism from the outside, casting upon it the Hellenic philosopher’s priority of knowledge rather than morality.
Anyway, to answer the Rambam part of the question... Yes, the Rambam and Qabbalah's explanation of mitzvos contradict, but most of us would be demanding the burden of proof from the Rambam.