Why when trading gold apart of purity measurement its mass is always measured worldwide with a balance and not a scale?
A scale, such as a typical spring scale, measures weight. Its accuracy depends on the gravitational field as well as on the stiffness of the spring. The gravitational field can vary up to 0.7% with position on Earth's surface, and stiffness can change with temperature. On the other hand, a balance, such as a typical double-pan balance, compares the gravitational force (weight) on the unknown mass to the gravitational force on some reference mass. Since the gravitational field is exactly the same on both masses, the balance actually compares the masses themselves. When very precise measurements are needed - such as gold trading or a chemistry experiment - a balance is much more reliable.
Both the question and much of the discussion assume an ideal distinction between balances and scales that does not reflect current practical definitions. Most modern high precision "balances" are not "balances" in the traditional sense. As one manufacturer puts it:
Today’s terminology would define a balance as a scale with a higher resolution.
If it is true that gold is always measured with a "balance" and not a "scale", it may just be because precision mass measurement instruments are commonly described as "balances", independent of their technology.
Modern analytical balances are complicated. They don't just hang weights in pans at opposite ends of a beam. High quality "scales" and "balances" both use load cells or tuning fork sensors, and (quoting the above manufacturer):
a scale normally consists of a load cell base, a display, A/D and CPU processor. A balance on the other hand may have a load cell, force restoration, or tuning-fork weighing mechanism, a display, processor, and display graduations in excess of 100,000.
Even if the sample is placed in a pan at the end of a beam, the force at the other end does not come from balancing weights, but instead from electromagnetic force restoration. (See, for example, Figure 2 of "Simplifying the electronic balance load cell".)
A load cell or tuning fork sensor measure force like a spring, so a modern high precision "balance" acts like a scale and is not insensitive to variations in the local gravitational field. For example, the reference manual for a best-selling Mettler Toledo JET1003G/00 Gold Balance explicitly states (in Section 4.6.5) that
To obtain accurate weighing results, the balance must be adjusted to match the gravitational acceleration at its location.
Other manufacturer's make similar statements about their analytical balances, e.g. Tovatech
Next you must calibrate the balance to compensate for gravitational effects at its location.
that consists of a motor and one or more weights of known mass that are housed within the balance itself. A user may initiate a process where the internal weight is automatically applied to the load cell and the measured value is checked against an expected value. If these values are different, the balance will adjust itself to produce correct measured values for the given load.
I'll bet this is because someone who is watching a jeweler, etc. weigh one of their possessions wants to see the weight on the other side of the balance as a demonstration that the weighing device is not rigged.
My bet is wrong, however! see Markoul11's commentary below. Now, why would someone like me for instance confuse weight with mass? this is because I am a recovering ex-engineer, and I have not yet mastered the part of the 12-step program for ex-engineers that teaches us that mass and weight are not the same thing. Please grant me the serenity...
...but I'll leave the rest of my answer here for your entertainment, not your edification.
This effect is not limited to weighing gold. At a time when precision electric metering pumps did not exist, a gas station operator would first pump gas out of an underground tank up into a tall glass cylinder at the top of the pump that was marked in gallons, so the car owner could see that there were ten gallons ready to dispense, and then he would release the gas to flow down into the car's tank. No cheating possible!
But when gas pumps went electric, the pump manufacturers installed little glass viewing bulbs on the side of the pump, which contained a plastic turbine wheel which would spin when gas was flowing out of the pump and into the gas tank. This told the car owner that gas was indeed flowing while the pump was running (and adding up the purchase price).
Then pumps went digital, and as there were no crooked gas station operators left at that point trying to earn their cigarette money by boogering gas pumps to cheat their customers, those flow indicators went away.
A balance will show the same reading for an object even if you go to the moon.
Where $g$ is the planetary body gravitational acceleration constant at which you are taking the measurement. A balance will correctly show the same mass of the object interdependent of where you are the moon or the Earth. Using a spring scale on the moon will report the gold to weight about 1/6th of the weight on Earth.
Balance measures invariant rest mass m, scales measure weight which gravitational acceleration $g=9.81 m/s^2$ varies on Earth per latitude due to centrifugal forces effectively (Earth's spin) changing $g$.
So, due to spin of the Earth and centrifugal forces created, the weight of an object can vary between the poles and Equator of the Earth depending latitude position up to 0.3%. Actually, at the Equator the Earth spins at about $1600 Km/h$ and an object there will weight about -0.3% less than at the poles due to the spin of the Earth. Usually this is not a problem but for a relative large quantity of gold this can make quite a lot of difference in money value.
Finally, calibration weights for a scale don't solve the problem since they are subject of the same centrifugal forces. Unless, the digital scale is programmed to input latitude position or have another way to compensate they will have always this drawback.
Of course a solution would be to input manually the nominal value of a calibration weight put on the scale even if it reads different in the scale per location latitude you are currently and hope that the nominal value stated for the mass of the calibration weight (possible measured with a balance) is accurate. So for example if a calibration weight says it is 500g, you calibrate your digital scale if there is an option via manual typeset input to 500g even if it reads out for example 498.44 at your location. However, all other possible systematic errors of scales still apply.
In general, the most accurate measurements are not made using a measurement device that produce a reading within a range of values, but with a device that simply measures whether two things are the same or not used with an accurately made reference.
It's much easier to make a very accurate reference for a single value and a device that detects whether two things are the same or not.
For mass this would be a balance with reference masses. For length measurements this would be a variance dial indicator with gage blocks. The balance or indicator might a readout to show how far off the workpiece is from the gage block, but the most accurate way is to arrange your references to get the same reading on both the references and on your workpiece.
Because while a balance normally compares an item against one or more weights (which are handled as items of considerable value in themselves), it can also be used to compare two sets of weights against each other at short notice.
What is more, because of the balance's simplicity and its removable pans it can always be checked to be fair, and that fairness will be independent of temperature etc. and, within limits, the range of weights used for the test.
So if a customer took (for example) a ring to jeweler A who appraised it as 140 grains and then to jeweler B who appraised it as 150, it would be likely that at some point they would be "encouraged" to compare their weights against each other: which could be done fairly on either their own or somebody else's balances.