In regards to German direct injection. There are three basic types of fuel injection
1 Throttle body injection where a single spray injector puts fuel where a venturi type carburettor would normally be. This allows more precise control, provides vaporisation from the high pressure of the spray and has no negative G issues. The Allied engines tended to move from carburettors to this type of injection though they were often misnamed carburettors.
2 Multipoint injection into the valve intake plenum, the injector sprays on to the back of the intake valve rather than direct into the cylinder. This is the kind most often seen in cars up to about 2010 or so.
3 Direct Injection. This is the technical high end, it involves direct injection into the cylinder. This is what the Germans used. As shortround6 says this was mechanical but mechanical systems are no less precise than electrical. Modern engines can use very advanced techniques to put in multiple squirts of fuel to stir up the mixture and create stratified charge (rich near the spark plug to get good ignition and lean elsewhere to get good economy). The high pressure creates excellent vaporisation.
Now as to why. Anthony Kay in "Junkers aircraft and their engines" says that American and British companies had so tied up carburettor patents the Germans had no choice anyway.
Robert Bosch had developed direct injection to essentially invent the high speed diesel. Up until then Rudolf Diesels invention relied on compressed air injection. It worked well but was rather bulky. The expertise developed by German companies such as Robert Bosh, Prosper L'Orange and Junkers was merely applied to petrol engines.
Now as to the advantages and disadvantages.
1 Good Cold starting. During the Stalingrad airlift the most effective transport was the He 111 because its engines would start reliably. The Ju 52 needed elaborate heated ducts to each engine and often couldnt start.
2 No danger of fuel fractionating in the ducts (as happened to the P38) and no issues with getting vaprisation of fuels that dont meet spec.
3 The biggest advantage was that direct injection allowed large valve overlaps so that intake and exhaust was open simultaneously. This allowed the end gases to be effectively scavenged and swept away all of the dead exhaust that didn't produce any power and could cause preignition. The DB601 also used a tuned resonance effect like a trumpet, to resonance scavenge the exhaust. On the DB601E onwards this was so large that the engine idled poorly and variable length intake ducts were introduced. Variable intake ducts were not feasible on a radial so BMW were planning variable valve timing on the BMW 802. Do this on a throttle bodied injection engine like a merlin you would loose a bit of fuel.
Now the disadvantage.
The British and latter American engines used aromatic fuel like 100/130 as a charge precooling mixture sprayed in before the supercharger. This dramatically contracted the air and increased charge density and reduced supercharger work.
In about 1942 the Germans improved their green dyed C3 fuel from about 94/115 to 96/125 then 100/130. Towards the end of 1942/early 1943 they introduced 'rich mixture injection' on the BMW 801D2 of Fw 190F ground attack aircraft. Essentially they must have turned down the direct injection and injected the majority of fuel into the eye of the supercharger as on the Merlin. Fw 190A fighters received only 'increased boost' which just increased boost settings to take advantage of the better fuel. Latter the two systems were combined.
The BMW 801 didn't really have large valve overlaps as it was a radial. However the DB605A did so it would loose a lot of fuel if rich mixture injection was used, I think this is why water injection was preferred, its not such a tragedy to waste water as as to waste fuel.