Usually, the term "Syriac Christian" refers to several different denominations that share Syriac as a liturgical language, but these groups had rather different experiences of the Ottoman period. I'm only going to answer for the one group with which I am most familiar, but I'd like to add some background to explain why they should be differentiated.
As a quick run-down, the Syriac churches have two broad divisions: the Eastern - itself divided into the Church of the East (sometimes called the "Nestorian Church") and Chaldean Catholic Church; the Western, which the Syriac Orthodox Church (no relation to Eastern Orthodoxy), which also has its own Catholic offshoot, and the Maronite Church, which is also a Catholic church.
A problem arises when attempting to pin down an ethnic identity, something which is incredibly contentious within these communities to the present day. While the majority consider themselves Assyrians, a minority of Chaldeans actually believe themselves to be the descendants of the ancient Chaldeans, while a significant minority of the Western branch believe they are the descendants of the ancient Arameans.(*) In other words, the Syriac Churches share a liturgical language but are very distinct theologically and not everyone agrees that they have a common ethnic ancestry.
And just to complicate things a little more when one speaks of "Syriacs" (in noun form), it is often understood to refer to the Syriac Orthodox and Catholics rather than the other groups. Why this all matters is that when someone asks an innocuous question about "Syriac Christians", we need to talk about 1500 years of theological division as well as a highly-emotional modern national identity discourse.
My interest is in the Syriac Orthodox and subsequent Catholic/Protestant branches (henceforth collectively "Syriacs"), and thanks to theology, geography and Ottoman political systems, there was only one relationship that really mattered: the one they had with Armenians. And this relationship was ... complex.
Under the Ottoman administrative system as it had evolved by the nineteenth century, the Syriacs were administered as though they were members of Armenian community (millet). This meant that the Armenian Patriarchate in Istanbul had considerable authority over their day-to-day lives and put the Syriacs in the unenviable position of not having direct representation with the Ottoman government. This isn't totally dissimilar to how, say, Arab Orthodox and Bulgarian Ottomans were both administered through the Greek Patriarchate in Istanbul, and like the Arabs and Bulgarians, the Syriacs in the 19th century had begun to agitate for their own independent millet to be recognised by the Ottoman government.
Unfortunately, the Syriacs lived far from centres of power and were furthermore a demographic minority everywhere they lived, usually outnumbered by Kurds or Armenians if not both (the one notable exception being the district of Midyat). To address this, a permanent Syriac Orthodox Church was built in Istanbul in the 1880s, and after petitioning the government, a Syriac Orthodox millet was ultimately recognised. A metropolitan was now representing the patriarchate's interests in the capital while the Patriarch maintained his traditional residence in the district of Mardin.
It wasn't an easy separation, however. In terms of doctrine, Armenian and Syriac Orthodox Christianity have much in common (the Armenian church is considered to be in communion with the Syriac Orthodox Church while the Eastern Syriac Christian churches are not), and they had a shared sense of ownership over many church properties. Disputes between the two over these were not new but they were exacerbated by the split and many of these would end up in the Ottoman courts for a resolution. Syriac accounts also record how the two groups sometimes came to blows over certain ancient monasteries along the Euphrates during religious festivities, even resulting in the Ottoman government dispatching soldiers to keep the peace. In 1911, the journalist Naum Faik chastised a Syriac priest for allowing Armenians to continue worshiping in a Syriac church in Mardin and appealed for the clergymen to assert themselves now that the community had "awakened" and achieved its independence.
This simmering dispute fits into a much wider context of Faik and other Syriac intellectuals' seeing their community threatened by the dominant Armenian language and culture in many of the places that they lived. Many Syriacs had already lost the Syriac language in favour of another language - whether it be Armenian, Arabic or Kurdish - which was lamented as symbolic of the decline of the community. Take for example, Aşur Yusuf, founder in 1908 of the first newspaper for the Ottoman Syriac community. Despite publishing articles on the revival of Syriac (even comparing it to the activities of the Jewish immigrants to Ottoman Palestine), he did not speak the language himself and in fact he took this message to his readers in the Ottoman Turkish language with the one concession that he wrote it using the Syriac alphabet. But to reinforce the complexities of the relationships here, even while decrying the loss of his community to an Armenian identity, Aşur Yusuf taught Armenian language at the American mission school in Harput and was married to an Armenian ... his daughter her father's biography in Armenian.
Let's consider Aşur Yusuf a little further since he was more than just a Syriac advocating for a language he could not speak. He was Protestant but was highly critical of the Syriac Orthodox Patriarch Abdallah for spending too much time in India and Jerusalem rather than "at home" working for the Syriac community as their main representative with the Ottoman government. Contrast with the previous Patriarch, Abdulmesih, who had called out the Armenians for historically trying to Armenianise the Syriacs, and even made some dubious accusations of them attacking and killing Syriacs during the Hamidian massacres of the 1890s. Other intellectuals used Assyrian and Aramean history to prove the ancient lineage of the Syriacs and how they were in fact an older Christian community than the Armenians or that their Partriarch (the titular Patriarch of Antioch) was on par with the Pope, while the Armenian Patriarch was merely a bishop in Istanbul.
The attempts to build confidence in an Syriac identity was by no means always successful, however, and there remained the perennial problem of Syriacs "conversion" to an Armenian identity. Take for example the case of a village priest near the town of Adıyaman, whose telegraphed the metropolitan of Harput to declare that he and his entire congregation were "becoming Armenian". Most cases of identity change were not quite as dramatic as this but whether through organic assimilation or marriage, it was always a threat: the records of post-war Syriac survivors, many are listed as having "become Armenian" in Hama or Aleppo or Beirut or wherever they fled to.
Politically, the Syriacs never had a deputy in the Ottoman parliament, whereas Armenians had many (incidentally, there was a Chaldean deputy from Mosul) and there was a debate in the community whether the Syriacs should seek to support more Armenian representation or whether to attempt to secure one for themselves. A respected Syriac doctor who served in the Ottoman military during both the Balkan Wars and on the Eastern front during WWI argued the Armenian position while Aşur Yusuf called for cooperation between Syriacs, Chaldeans and "Nestorians" (his words) to gain as many deputies as the Armenians since they had "a combined population of two million" as well.
Of course we know that the story ends with the Ottoman government completely failing to uphold the promises it had made to its non-Muslim populations following the 1908 revolution. Aşur Yusuf and others saw a unified Assyrian identity as being able to strengthen their position vis-a-vis Armenians and looked to the Ottoman state and constitution as protective of Syriac interests. As a footnote, it's interesting to see what changed with the Armenian genocide and the declaration of the Turkish republic: Armenians became a recognised minority in Turkey while the Syriacs did not; the status of Syriac church properties stayed in the courts but this time with the Turkish state as the opponent; on the other hand the Syriac community finally had its first parliamentary representative, winning a seat as part of a primarily-Kurdish party.
(*) This is skipping over the large populations in India who adhere to various forms of Syriac Christianity. It's also glossing over the fact that many Syriac Christians of all stripes began converting to Protestantism in the late Ottoman period.