Without something else to work off of, no one will be able to assist you.
Size difference ~may~ be an indication of the design.
Usually the wall warts are defined by the following:
Regulated or unregulated.
Linear power supply or switching power supply.
The cord end type
The polarity of the cord end.
So, you are saying that the voltage output (6V) and current capacity (300ma) are the same.
One is twice the size (roughly) of the other.
If they both put out 6 volts at up to 300ma, but one is regulated and one is not, that could be your difference. Way to check would be to test the voltage under load and with no load. A regulated power supply should stay pretty level, voltage wise. An unregulated power supply may have a higher voltage when not under load, and a lower voltage depending on the load.
Switch mode vs linear power supply would be hard to determine without opening it up.
A switch mode power supply usually will accept a much wider voltage range, like 90 volts AC to 240 volts AC, and it'll say that on the enclosure.
A linear power supply will only work properly at one voltage, in your case, 120vac.
Again, photos would be worth a thousand words...