They are Globally unrelated. max_connect_errors is one of Server System Variables , it is a per host basis counter while Aborted_clients is one of the Server status variables which is a global information counter for all clients/hosts.
Another reason they are not related, when the host whose max_connect_errors counter is in an incrementing cycle due to connect errors but then that host establishes a successful connection, the error count max_connect_errors for the host is cleared!
max_connect_errors is incremented for a particular host when the host fails to establish and no successful connection (threshold results in blocking the host). It happens when the handshake with the server was interrupted. If it wasn’t interrupted, it counts as “success” and reset the host counter – regardless of whether the end result was a successful connection or not. so it can be considered as a network performance counter, Note that it does not even strongly stand for security issues. You can test this by telnet MyServer 3306 then pressing CTRL C instead of proceeding ..
This counter could be cleared with mysqladmin flush-hosts; as in this post.
On the other hand, if a client successfully connects , but later disconnects improperly or is terminated, the server increments the Aborted_clients counter.
This can be caused by many things - The client exited without calling mysql_close(). Client connection exceeded wait_timeout without interacting with server. The client connection wast cut off like when turning off the PC.
Server status variables provide information about Server operation, It also includes Aborted_connects which is just a statistic for DBAs- not used by mysqld to determine server behavior.