As Johnny was saying in small piston airplanes you always take off with maximum available power. The reasons why get a bit deep & I’ll let the light plane folks explain if they can. I can come back to it later if nobody is game.
In airline jets the situation is quite different. We rarely use maximum power to take off. Instead we tailor the power to the minimum necessary (defined below) to hoist the weight we have from the runway we’re using under the atmospheric conditions present.
There are a host of factors which affect the takeoff power requirement. We have a whole department that does nothing but evaluate runways all over the world and maintain the data tables which drive these decisions.
Ultimately we need to be able to:
- Take off normally and climb steeply enough to not hit any obstacles near the airport.
- Start a normal takeoff, have an engine fail early in the takeoff roll, and get stopped before we run off the end of the runway.
- Start a normal takeoff, have an engine fail late in the takeoff roll after it’s too late to stop, but still get airborne by the end of the runway and also still climb well enough to avoid the obstacles near the airport. As well as perhaps thread our way between mountains out a canyon or pass.
You can clearly see how a lower thrust setting makes each of those maneuvers have less margin versus a higher power setting.
You can also see where a headwind, a cold day, a high atmospheric pressure, a sea-level airport, a long runway, a level or downsloping runway, flat nearby terrain with no obstacles, or a light load would be helpful.
Conversely a tailwind, a hot day, a low atmospheric pressure, a high altitude airport, a short runway, an uphill runway, nearby hills, mountains, buildings, antennas, etc, a heavy load, or a wet or snowy runway would be unhelpful.
So how do we decide? …
In effect we iterate a calculation backwards starting with current conditions and full power and determine whether we can accomplish all 3 takeoff criteria safely. If not, it’s time for another long-winded post by me. 
If we can get off the ground under maximum power, we back off the power a smidgen & recalculate. Iterate until we first can’t make one of the 3 success criteria. Then add back a smidgen of power & we’re good to go.
As a practical matter all of that is pre-computed so the manual process is done by consulting a data table for the particular airport & runway and entering it with the various weather variables to determine what power setting meets the most restrictive of the 3 safety criteria.
Typically this is all handled automatically by our HQ computer systems, and we’re responsible to ensure the weather inputs used match current reality, verify the reasonableness of the results, ensure the settings are correctly transferred into the aircraft, and then bet all our lives on it actually being correct.
And yes, there are safety margins built into the three takeoff criteria. But less than you might imagine.
To answer the questions somebody is sure to ask: Why use less than full thrust if it represents an erosion in performance and in some sense an erosion of safety margin?
The answer is three-fold. 1) Less than full thrust is much quieter for the neighbors. Noise regulations have real teeth worldwide. Billions of dollars of perfectly good airplanes have been scrapped because they’re noisier than folks living near airports are willing to live with. 2) Less than full thrust makes engines last longer, saving money. Which means lower fares, which makes customers happier. 3) Engines almost never fail mechanically except at maximum thrust which is maximum thermal, centrifugal, vibrational, etc., stress. About 98% of the industry-wide “kablooey” engine failures happen on < 10% of takeoffs done at full thrust. Kablooey’s are real rare. By avoiding max thrust we make them much rarer yet.