I know its pretty old but I just encounter the problem and there is what I saw in the SQL doc :
[For best results when using BETWEEN with date or time values,] use CAST() to explicitly convert the values to the desired data type. Examples: If you compare a DATETIME to two DATE values, convert the DATE values to DATETIME values. If you use a string constant such as '2001-1-1' in a comparison to a DATE, cast the string to a DATE.
I assume it's better to use STR_TO_DATE since they took the time to make a function just for that and also the fact that i found this in the BETWEEN doc...