I get the sense you are most interested in unusual linguistic features of Latin, which I'm not qualified to talk about.
It's also worth noting that 'why study Latin' is a well-addressed question in general (often aimed at the perspective of high school or college students choosing courses); Googling it will get you lots of results.
But besides this, a few suggestions in no particular order:
Access to a wide body of academically- and culturally-significant literature and writing
Heroic epics, mythology, biting satire, carefully-crafted epigrams, political speeches, histories, philosophy, natural science and much more; all of these are not just present in the body of Latin literature but form a canon of work which has influenced Western and global artistic output ever since. Reading these in the original gives you a fuller appreciation for the craftsmanship and artistry of the work, as well as avoiding the barrier than any act of translation puts between the reader and author.
Beyond this, I just think it's extremely cool to be able to read something that a person wrote thousands of years ago.
Not to mention, people kept writing in Latin for hundreds more years! Latin was the language of scholarship until comparatively recently. If you want to read about mediaeval academia you'll need Latin.
Access to the study of ancient Rome, one of history's major empires
The Romans conquered and governed a huge part of the ancient world, which is both interesting in itself but also has lasting consequences and lessons for the West and wider world. Only the other day I was reading an article about the lessons that a careful study of the Roman army when it became a volunteer force has for the US military today.
Insight into the Romance languages, and the Latinate words in English
A solid grasp of Latin acts as a key which can unlock, to varying degrees, languages like Italian, Spanish, Romanian, French and more. Together with Greek, the bulk of technical English vocabulary is derived from Latin and having a grasp of Latin enables you to parse vocabulary which might otherwise need to be looked up in a dictionary.
Experience with the language that was historically considered to be a grammatical paradigm
As you sound like you're into linguistics, you've probably come across terminology that seemed odd or categories that were a poor fit for a language; this is often because grammarians based their thinking around Latin. Knowing Latin can help understand where they were coming from.
For better or for worse, among Westerners Latin is associated with being cultured, sophisticated and knowledgable. If you are cynical, think of this as a fundamentally empty signalling exercise; it could nevertheless benefit you to have that signal. More generously, this is about participation in a scholarly tradition and shared body of knowledge (for the kinds of reasons above).