Having studied both, I can say that there are basically no contradictions between the two. However, there is a difference in the approach. I would recommend watching this lecture which specifically sets out to answer your question.
To summarise (far too briefly to do it justice), they are both seminal works of their authors, who both intended that this particular path will bring about the redemption. The "difference" in approach is that Mesilat Yisharim seeks to elevate the human being to perfection, whereas Tanya seeks to bring Godliness down into the world (and there is a huge amount of overlap). Both of these tasks are compatible and are both necessary to finish the job.
A mashal I like to employ is that Mesilat Yisharim is a "how to be a good husband" manual. It's goal is to refine and elevate you. Tanya is more of a work specialising "getting to know your Wife". As you can see, both are very important jobs to engage in in order to achieve a successful and intimate marriage.
R' Loewenthal does make a point that he believes is a key difference. Mesilat Yisharim doesn't devote much if any of its words to struggle. It's simply a very long a to b of "do this, and when you've mastered it, do that, and so on". Tanya on the other hand devotes a huge amount of work to explaining that some tasks in life are endless. The Benoni is always going to struggle with temptation, and it's not reasonable to expect this aspect of the battle is ever going to be winnable. The Tanya explains that the purpose of this is that the struggle itself is valuable, and brings down Kedusha and Godliness to the world.
This is why the lecture is called "Competing Ideals of Ascent and Struggle"
However, I personally prefer to focus on the similarities! The "Naki" is Ramchal's "Benoni", and so much of what each says about this type of person is identical. What we learn about in Mesilat Yisharim's chapters on Perishut, Tahara and Chassidut are ideas that are very much present in Tanya. The Kadosh and the 'Tanya Tzaddik' are very related. It's amazing how similar these works are, and how clear it is that we have a strong mesora that each Gadol is able to paint and weave into each their own incredibly holy work that has profoundly affected all of us to this day. The differences are mainly differences in personality and approach.
Rabbi Nissan Mendel, quotes the compilers foreword from the Alter Rebbe to his magnus opus, the Tanya. He explains that Tanya is:
[...] essentially a work on Jewish religious ethics. The author is primarily concerned with the forces of good and evil in human nature and in the surrounding world, and his objective, as already pointed out, is to pave a new way to achieving the ultimate purpose of creation.
Messilas Yesharim on the other hand, has the goal of explaining to us how to achieve perfection in our divine service and the necessity for its purity and cleanliness
In the introduction, it says:
How will we purify our thoughts if we don't exert ourselves to cleanse it from the imperfections instilled in them by physical nature?! And what of all our character traits, who likewise are in need of so much rectification and correction. Who will correct them and who will rectify them if we do not attend to them and are not exceedingly meticulous in this?!
to teach myself and to remind others the conditions of perfect service according to their proper levels.
The Tanya on the other hand explains things like "the psychological structure of the Jewish personality", e.g. that we have an G-dly and a animal soul within us. The Tanya also speaks on the subject of the inevitable conflict ensuing from the two divergent sources of consciousness and how to channel them.
The Alter Rebbe says in the compiler's foreword that he wrote the Tanya to give "counsel" to those that seek the path to G-d. Every people has struggles in his Avodas Hashem. People want an private audience with their Rebbe, so the Alter Rebbe wrote the Tanya as an private-audience so to speak for those who seek to perfect their Avodas Hashem.
Mesilat Yasharim teaches the path of Nevuah, the path of prophecy, meaning how to prepare oneself to become one of the B'nai Aliyah.
Like its author says in general, he is teaching the analog to the allegory found in the teachings of The Ari z"l. So too, Mesilat Yasharim follows the teachings of the Ari z"l that are found within the writings of Rabbi Chaim Vital, z"l in Sha'ar Ruach HaKodesh, in particular the 4th section.
This is generally the domain of the Yechidei Segulah, the select, distinguished few within each generation. It is based upon the teachings of Rabbi Pinchas ben Yair, z"l, whose teachings can also be found in Midrash Tadshe.
The Tanya is comprised of 4 main sections, Likkutei Amarim, Sha'ar HaYichud, Igeret HaTeshuva and Igeret HaKodesh. The 5th section, Kuntress Acharon, is primarily an explanation of Igeret HaKodesh. It is suggested by some that a person new to the Tanya should begin at the second section, Sha'ar HaYichud which deals with the subject of the Shema, then proceed to Igeret HaTeshuva and only then begin Likkutei Amarim and the balance of the Tanya.
Like the title page explains, Tanya teaches the path of the Benoni, the intermediate person, who is neither completely righteous, nor r"l, the opposite. This means that it is intended for the majority of people.
It's purpose is to teach how the majority of people are to serve G-d and to reveal that G-d dwells everywhere, including in this physical, material plane of existence.