The relation to Arduino is small, because you connect to its pins. However, this issue is an electrical engineering issue at the end of the day.
Static drive vs multiplexed drive
If you use a GND connection on pin 1 and drive pin 6 statically HIGH, the LEDs will be on all the time.
In contrast, the digits are multiplexed and are on only during their time slot, and off during the other time. This reduces the effective brightness roughly by the number of time slots.
Calculating the resistor
Your calculation "5 V divided by 25 mA gives 200 Ω" is correct by Ohm's law. But your assumption of 5 V at the resistor is wrong.
Each LED has a "forward voltage" that depends on the specific model, roughly on its color and technology. The data sheet tells us 4.0 V typically at 20 mA (for the digits, as they have 2 LEDs in series). It raises with the current, but not linearly, to about 4.05 V at 25 mA. Please be aware that these are typical values, the concrete values can differ a lot!
The dots' LEDs are single LEDs, you can assume a forward voltage of about 2 V for them. Take this into account when you calculate the resistor(s).
Assumed that the Arduino's pin would output 0 V for LOW and 5 V for HIGH, which is idealized, the remaining voltage at the resistor is about 1 V. Divided by 25 mA results in a resistance of 40 Ω.
Now the multiplexing comes into play. Since you use 4 digits, most probably each of them is driven roughly during 1/4 of the time. To have the average of 25 mA, you need to multiply the current by 4, giving 100 mA. The resistor will have 10 Ω then.
But the display does not allow this according to the data sheet. Unfortunately it does not say anything about a duty cycle of 1/4, only 1/10. If you accept potentially reduced life time, you can drive it with 100 mA nevertheless. If you don't, reduce the current.
Driving capabilities of the Uno's pins
When multiple segments are on per digit, the HIGH side of each segment still carries 100 mA as calculated above. But on the LOW side (the common cathode) the current is the sum of all driven segments, up to 7. Outputting an "8" will result in 700 mA!
The pins of your Uno cannot source or sink a lot of current. If you try to do more, real world effects according to physics jump in and make the mental model invalid. According to this answer on SE/EE, the maximum current per pin is 40 mA.
If the current is higher, a LOW will not hold the voltage near GND. It will be significantly higher. The same is true for HIGH. The data sheet of your Uno's MCU will tell you more, look for "pin driver strength".
This reduces the usable voltage on resistor and LEDs. This leads to less current, until some balance point is reached. Please be aware that you are using the device outside the specifications here, which could damage it permanently.
Therefore you will want to use driver stages (ICs or transistors) to be able to use more current. But this design is not clearly not Arduino-related. Please do your own research.
So is there a voltage difference or current difference?
... neither ... it is the ON time difference