Your primary voltage is of no interest, it is as you say transformed down to 6V secondary.
The capacity in time is measured in AH, the 4.5AH battery will in theory keep up for 4.5 hours if you draw 1A out of it and 9 hours if you draw 0.5A.
There is also a max rating for the top current you can draw from the battery, 0.36A versus 1.2A for the other. The difference between them is 333%...
Top current and capacity is dependant on the area size of the lead plates so I guess the 1.2A battery is quite a bit bigger than the old one unless the mfgr has found some tricky way to organize the lead plates and packing more of them into the same size battery
Since the new battery has a bigger lead plate area, can give out a higher current, then it will also require a higher charging current and this is where it can go wrong for you.
Modern electronics does rarely have reserves, components in the charger are prolly not rated higher than needed for charging your old battery. The new battery will draw more from the charger so there is a risk that it will give out black smoke...