I initially wondered why the antagonistic aliens decided to attack Earth on 4th of July since it seemed to be a rather big coincidence in the second movie. The answer is actually pretty simple. They wanted to demoralize the human inhabitants by basically curbstomping their defenses on the very same day they celebrate their victory over the first invasion. Plus, security would probably be a bit more lax on that time of celebration which is a good bonus.
Also, the planet is in (roughly) the same orbital position, allowing a more-or-less identical approach (presuming you're able to avoid the outer planets). If they'd attacked on, say, January 4 then the sun would have been in the way, potentially requiring a bit more moving within the solar system (presuming a roughly-on-the-elliptic approach).
Charlie pretty much confirms this theory in the novelization, pointing out that the position of Earth was identical in previous alien intrusions (The War of 86, Roswell, etc.).
Ever wonder why the original Mothership remained undetected by any Human means until it was in close proximity to the Moon (there is an entry in Artistic License Physics discussing it)? When the Harvester Ship shows up, it is revealed that it possesses a cloaking device, which kept the Harvester Ship hidden until it was approaching the Moon. This is how the original Mothership could escape detection from telescopes and the like, and how it got so close before anyone noticed (gravitational effects aside).
Several of the aircraft used by the EDF are obviously Real Life 20th century aircraft with their engines swapped out for angular futuristic designs. The examples we see are pretty much bigger support aircraft or bombers. This makes sense for a couple of reasons: After the War of 96, the first order of business would be upgrading their surving forces with the newfound technology, and most of the surviving aircraft after that war were probably the aforementioned larger support craft and bombers which wouldn't have been thrown into the pitched aerial battles that defined the first few days of the war. Finally, swapping out older engines for newer and more effective designs is a common upgrade for older aircraft such as the E-3 Sentry.
One would wonder why the Harvesters would go to the trouble of harvesting the Earth's molten core to help fuel their technology and ships, since one would assume that there were would be more viable resources to tap into than a planet's core. However, the Earth's core consists of the molten outer core, and the solid inner core, and both are believed to be comprised of mainly iron and nickel. This means that a planet's core is effectively the richest source of metals for a planet, perfect for a vast space faring race with ships as vast as the Harvester ships. It is probably the means by which they build Harvester, Mother, and City Destroyer ships in the first place, alongside the organic components.
That's frankly one of the (many) places where this movie lost me. If all they want is the core, why aren't they harvesting uninhabited planets that don't put up a fight? There's quite a few of them to go around. It made far more sense in the original that they wanted Earth for its Earthiness. Air, food, water... y'know, habitable biosphere for their civilization to use until they stripped it bare and moved on again.
At the very beginning of Resurgence we actually do see the Harvester ship that would eventually attack Earth stripping a barren rock planet. It is possible that Harvesters are the ships that come in after a planet has been wiped clean on the surface, of resistance and resources. But they made a special case for destroying Earth outright out of pure spite.
Well, Earth's molten core is highly useful for the organic life on the planet (it provides the magnetic field that shields us from all kinds of harmful radiation from space). Planets that don't have molten cores (like Mars) don't have a magnetic field, and much more harmful radiation hits the surface. Perhaps most planets with molten cores are habitable ones, and the molten cores are best for what the Harvesters need?
This might be more Fridge Brilliance than Horror, as Harvester's end goal is to destroy all life on Earth and taking out the core would be a pretty efficient way of doing that.
At the end of the film, the salvage team cheers that they're rich, because they'll be paid $100 million they were promised by the UN. Even though half of Asia was just dumped on Europe. Good luck having all that money be worth anything.
While the immediate damage to earth was indeed catastrophic, humanity is about to enjoy a new renaissance thanks to the massive technological knowledge that the alien sphere is going to share with us. They will be fine and very rich.