Whats the difference between a battery charger & Transformer to power a 12 V 3 A bilge pump?
1) Why does my 20 year old 4amp 12volt Battery charger (see image) work fine as a power supply for the 3.8amp 12v pump?


Probably because the battery charger has different heavy-load behavior then your power supply. i.e. it might have a slow-blow fuse and also respond much slower to being overloaded.


  2) &amp; can I use the battery charger as a permanent power supply for the pump, ie could it overheat &amp; cause a fire?, as they are not normally left on permanently.


Regularly overloading any type of equipment is a bad idea. You are then starting to rely on the manufacturer's fail-safe implementation, if there is any!


  3) As I don't want to waste money buying another unsuitable transformer. Will a " 40amp DC12V Driver AC 230V Power Supply Transformer for LED Strip Light" see image be a suitable power supply for the 3A 12v pump? 


It should be sufficient for the job but then you can have very high currents running.  

That by itself is a dangerous situation.

e.g. Your 20 meter cables might have an resistance of about 0.3 ohm. If the far end is short circuited your supply would happily pump 40 Amps through the cable heating everything up.