I have used both - I replaced my delonghi with the gaggia quite recently.
They both make a reasonable cup of coffee, however the gaggia is a better machine.
Build quality, the gaggia wins hands down. It's a chunky mostly metal affair, probably twice the weight of the delonghi; the delonghi is somewhat plasticy. The portafilter for the gaggia is a big chunky brass affair, rather than the delonghi's plastic with steel insert. This means it holds heat a lot better so tends to give a more consistent cup of coffee.
The gaggia has a much better water flow too - you get a more even delivery of water and it has a pressure relief valve so when you stop the pump it de-pressurises the portafilter. This means the coffee stops immediately. With the delonghi the coffee continues to dribble out after the pump is shut off making judging a cup full to be a bit guesswork.
I also found that the way the filter basket is designed with rubber seals in the delonghi wasn't great - the seals had started to perish on mine after about 18 months of use. The gaggia has no rubber seals there so it's not an issue.
I mainly drink espresso so I don't have much comment on the steam wands - I've not even used the gaggia one yet. The delonghi one frothed milk up competently enough though.
Also the actual reason I replaced the delonghi is the boiler thermostat had gone a bit wonky - instead of holding the water at the correct brewing temperature, it would heat it up to boiling point, regardless of having steam or coffee selected. This meant I was ending up with burnt coffee more often than not.
So it was good while it lasted, but 18 months of more or less daily use killed it. Not great for something that cost over £100.