The first is natural English. The second is poor grammar unless they're not even talking about the river. They could be using slang / poor grammar intentionally, to speak less words; or they could be talking about a company or location that's named after the river.

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."

I can't think of a context in which "down to Mississippi River" makes sense, but I'll let you know if I can find one.