Can “tar” backup incrementally

I would not recommend using FTP for any files you care about (particularly database backups if that database contains customer information), but that is ultimately up to you.

To start with, script the creation of your backups on the Linux server with something like the following:

#!/bin/bash
# 
# Call script with:
# /cron-mysql_bkup.sh mysql_hostname mysql_username mysql_password /backup/directory
# 
if [ -n "$1" ]; then
 MYSQL_HOST="$1"
else
 echo "MYSQL_HOST not specified."
 exit 1
fi
if [ -n "$2" ]; then
 MYSQL_USER="$2"
else
 echo "MYSQL_USER not specified."
 exit 1
fi
if [ -n "$3" ]; then
 MYSQL_PASS="$3"
else
 echo "MYSQL_PASS not specified."
 exit 1
fi
if [ -n "$4" ]; then
 DIR_BKUP="$4"
 mkdir -p ${DIR_BKUP}
else
 echo "DIR_BKUP not specified."
 exit 1
fi
# 
# Generate filename
# 
BACKUP_FILENAME=`date "+%Y-%m-%d"`".sql"
# 
# Dump all databases
# 
mysqldump -h${MYSQL_HOST} -u${MYSQL_USER} -p${MYSQL_PASS} --all-databases > ${DIR_BKUP}/${MYSQL_HOST}/${BACKUP_FILENAME}
# 
# Write status of the operation to the mysql.log file in the backup directory
# 
if [ $? -ne 0 ]; then
 echo `date`" - ERROR creating backups of host "${MYSQL_HOST} >> ${DIR_BKUP}/mysql.log
else
 echo `date`" - MySQL Backup complete for "${MYSQL_HOST} >> ${DIR_BKUP}/mysql.log
 BACKUP_TAR_NAME=${BACKUP_FILENAME}".tar.gz"
 tar czf ${DIR_BKUP}/${MYSQL_HOST}/${BACKUP_TAR_NAME} ${DIR_BKUP}/${MYSQL_HOST}/${BACKUP_FILENAME} && rm ${DIR_BKUP}/${MYSQL_HOST}/${BACKUP_FILENAME}
fi

This script will:

1. Check to see whether all the required parameters were set
2. Dump all MySQL databases to a file named YYYY-mm-dd.sql under the backup directory specified (this directory will be created if it does not exist)
3. GZIP the file and, if the GZIP operation succeeds, delete the original SQL file

This script should be fairly straightforward for you to customize if you wish to - see the Advanced Bash Scripting Guide to learn more about doing so.

... once your databases are backed up and can be found in a specific location, why not automate backup retrieval with WinSCP instead of transferring your files over FTP?

Have you considered using tar compare option to see which files are changed? This should give you a quick list of candidates.