You want to know if there are differences between the words "run" and "run across".
Well, the only way to find out is to run some sentences using both the "run" and "run across" words.
Helen ran across the street to meet her boyfriend. <-- O verbo run aqui é usado no passado
Marcos is going to run the São Silvestre in a few days. I just hope everything is going to be all right and he does not run across any obstacles!
Obs: The run across can also be used to express something different than "correr" in a sentence, as an example we have;
I was at the office Today and when I was running our accounting books, I run across some numbers that did not match with the total balance of our daily purchases. Maybe we should run these balances again to see if they match with the totals and avoid some terrible mistakes in the end of the Month.