The difference between the two phrases lies in the use of the definite article "the":
"Down to the Mississippi River": This phrase is grammatically correct and specifies a particular river, namely the Mississippi River. The definite article "the" is used here to indicate a specific, well-known river.
"Down to Mississippi River": This phrase is grammatically incorrect because it omits the definite article "the." In English, when referring to a specific, well-known geographical feature like a river, the definite article "the" is generally needed.
So, the correct phrase to use is "Down to the Mississippi River."