What you are seeing here is a phenomenon called a "flat adverb". You might think of it as an adverb without the -ly suffix (though not always.) "Slow" as a flat adverb and "slowly" as a regular adverb have exactly the same meaning.
Flat adverbs were once quite common in English but have fallen out of favor, and often people who are sensitive about grammar will complain about sentences like "please drive slow", or "you are doing good". As to whether such things are "right" or not I think it is hard to say, since English does not have some body that defines the language. What is "right" is what the educated and literate do with the language, and flat adverbs are still quite common, even though many would cringe at the aforementioned "you are doing good", most would not be too concerned with "please drive slow."
For sure there are some flat adverbs where there is no -ly alternative, for example, "Please drive fast" must be used instead of some putative "Please drive fastly", even though "Please drive quick" and "please drive quickly" are both common and widely accepted.
So the answer to your question: how to decide which to use where is simply that there is no rule, you can often substitute a flat adverb for a regular adverb, but you do run the risk of being criticized nit-pickingly, if you'll excuse such a word. Except in those few cases where a flat adverb is the only alternative. You can usually be safe by using the -ly, or equivalent, form, except in those few cases where such a word does not exist, or where there is a specific idiom. "Please sit tightly", "take it easily" or "drive straightly down the street" all sound pretty odd, for example.
Some more info on this at wikipedia or the ever enjoyable "Ask the editor" at Merriam Webster.