You could use a different set of symbols to represent the machine codes. But nobody bothers, because you wouldn't gain much.
ARM has an instruction called ADD. In ARM assembler, "ADD r0, r0, #1" represents the 4-bytes of machine code which constitute an instruction to increment register 0.
Whatever you call that instruction, you can't change the set of instructions available and still call it ARM assembler. It's still fundamentally the same programming language whether you call the ADD operation "ADD", or "SUM", or "PLUS", or "ADDITION". Since it's easier to use existing references if everyone uses the same names for everything, that's what happens.
One useful change might be to represent the instruction as "INC r0", since ARM doesn't have an INC instruction, and it's a common operation. This leads to macros in assembler languages. These genuinely do change the language, but once you have macros which emit multiple ARM instructions, you start to lose the WYSIWYG nature of assembly. Eventually you start to think that maybe you might as well just write C. I speak from experience (it wasn't ARM, but it was a macroised assembler).
One common difference is case - if you felt like being pedantic, you could argue that there are two different versions of ARM assembler language, one in uppercase and one in lowercase (or argue that there's one language, with multiple symbols for the same thing). Different disassemblers of the same machine code sometimes output different formats. Sometimes these are different enough that a particular assembler won't cope with all of them, or assemblers will offer their own conveniences which are incompatible with another assembler on the same platform. But really, it's all the same thing, and if you're bothering to draw the distinction, it's generally because you've been bitten in the ass rather than because anything good is happening...