I've usually had lovely weather in April in Italy, with of course some rain at times. But gloomy and overcast is NOT an accurate description, except for the odd day here and there.
As for temperature, I vividly remember that in Rome in September 1994, I was able to eat all my dinners outside (all the restaurants had outside tables, and all of central Rome was like a block party). However, in my several subsequent April visits to Rome, I ate all lunches outside, but dinners were inside as it wasn't warm enough. I mention this not only as a sign of the weather, but as something that made the visits feel different. I and a friend agreed that the ambiance created by everyone eating dinner outdoors was wonderful, but another friend didn't feel that way at all; he preferred the cooler weather of April and didn't feel he lost anything by not having dinners outdoors.
So, in summary, unless you care about eating dinner outdoors (and are willing to have hot days as a tradeoff), you will definitely like Rome better in April.