When people talk about different language, they mean more than just notation. For example, I can write "2 | 4" as "4 = 0 in Z/2Z". There is more than just a notational difference here.
A more non-trivial example: Say you want to say that 2 x 4 = 8. One way is to just write 2 x 4 = 8. Another way is to say that a finite set of cardinality 8 is the product of a finite set of cardinality 2 and cardinality 4 in the category of finite sets (in the sense that 8 satisfies the universal property of 2 x 4).
A more non-trivial example: Say you want to say that 2 x 4 = 8. One way is to just write 2 x 4 = 8. Another way is to say that a finite set of cardinality 8 is the product of a finite set of cardinality 2 and cardinality 4 in the category of finite sets (in the sense that 8 satisfies the universal property of 2 x 4).