If you're set on using Java Dates rather than, say, JodaTime, use a java.text.DateFormat to convert the string to a Date, then compare the two using .equals:
I almost forgot: You need to zero out the hours, minutes, seconds, and milliseconds on the current date before comparing them.  I used a Calendar object below to do it.
import java.text.DateFormat;
import java.util.Calendar;
import java.util.Date;
// Other code here
    String toDate;
    //toDate = "05/11/2010";
    // Value assigned to toDate somewhere in here
    DateFormat df = DateFormat.getDateInstance(DateFormat.SHORT);
    Calendar currDtCal = Calendar.getInstance();
    // Zero out the hour, minute, second, and millisecond
    currDtCal.set(Calendar.HOUR_OF_DAY, 0);
    currDtCal.set(Calendar.MINUTE, 0);
    currDtCal.set(Calendar.SECOND, 0);
    currDtCal.set(Calendar.MILLISECOND, 0);
    Date currDt = currDtCal.getTime();
    Date toDt;
    try {
        toDt = df.parse(toDate);
    } catch (ParseException e) {
        toDt = null;
        // Print some error message back to the user
    }
    if (currDt.equals(toDt)) {
        // They're the same date
    }