Whether a linguistic variety like Lowland Scots is a "separate language" or a "dialect" is entirely a matter of (often political) opinion. Often there's no sharp dividing line between the categories. Norwegian and Danish were generally considered to be the same until

The most diagnostic feature of distinct languages is the combination of vocabulary and grammar (lexicon and syntax, if you want to get fancy). Accent doesn't figure into it.

Scots obviously has plenty of words not known elsewhere - but I'm not sure that anyone has determined by how much they outnumber standard English words used in Scots - or if they do.

As for grammar, the grammar of Scots is almost identical with that of British (and Irish and American) English. Undoubtedly there are variations here and there, but while a non-Scots speaker will require a glossary to read Burns's Lowland poetry, he or she won't need a course in Scots grammar.

Contrast the situation between English and German, or English and Dutch, all of which were once dialects of a mutually intelligible West Germanic, even if it was some fifteen centuries ago. But they're now mutually unintelligible: a speaker of one can't even understand (much less converse naturally in) another without extensive experience or study, and preferably both. A Burns-style glossary won't be of much help.)

Unlike English vs. German, whether one thinks Scots is or isn't a "separate language" may well depend on where one is living, where one's forebears came from, or how one feels about political independence.