What is a raw data takeoff and is there a benefit to performing them?
Raw data means data read on the basic instruments. Not-raw data means enhanced information built from raw data and/or flight management system.

Flight director bars are not raw data, but a computation from basic air data, desired airspeed, roll angle limit, etc. At the next higher level, the autopilot can use the FD data to fly the aircraft. Similarly desired speed can be determined automatically and thrust can be managed by the auto-throttle.

Taking off and climbing with FD, auto-throttle and autopilot


Takeoff with FD and AP, just after liftoff (A320 simulator). Source: Youtube

The green bars of the FD shows what should be the attitude of the aircraft, SRS (speed reference system) guidance law has been used to determine the desired 165 kt speed. The pilot may fly the aircraft manually using these non-raw data.

In the present case the pilot has also activated another layer of assistance: Autopilot and auto-throttle. The AP follows the FD indications, and the thrust is managed by the auto-throttle so that the desired speed is achieved safely as soon as possible.

The indication in the top-left corner means autopilot, flight director and auto-throttle are currently selected / active.

T/O with raw data

Flying with raw data implies not using the FD or the SRS mode, nor the AP other modes or the auto-throttle.

The crew must determine the attitude to be used after the T/O rotation, and during the climb. They also need to manage the thrust required. For example, the roll angle has to be determined manually, depending on all factors discussed above and the desired horizontal track.


Takeoff without FD (A320). Source: Youtube  

Benefits

Not using computer assistance helps maintaining and developing pilot skills and prevents dependency on automation that can always fails. The correct attitude, speed and thrust depend on many limits  — some are displayed in the speed indicator band and the attitude indicator when they have video displays — that the pilots must have in mind.

So it's usual that pilots perform manual take-offs and landings and raw-data operations. The frequency of such permanent training may be influenced by the company rules (standard operating procedures, SOP) which can either encourage or discourage crews to use the airplane automatic features at certain phases of the flight.

On modern aircraft there are some mechanisms can't be inactivated as the direct transmission between controls and corresponding actuators have been replaced, partially or totally, by electrical links. The conversion between actual pilot movements and electrical command signals is subject to some level of filtering provided by the fly-by-wire system.

Other raw data operations


Raw data (ILS) approach.
Raw data navigation.


This post consistently uses Airbus-based explanations, as terminology varies between aircraft manufacturers (but principles are mostly the same for this topic).