How do Hilo and Kona compare, when visiting Hawaii?
They couldn't be more different. Each region has 2 distinct parts.
Hilo - north/ western / waterfront side. This region includes the area from the airport to the Belt road, continuing to Hilo bay (and inland). Because of two devastating tsunamis (1946 and 1960), which destroyed a huge swath of buildings, this area is basically stuck in the 1950s. I would describe it as "quaint", somewhat like a dying Midwestern town, except with a heavy Hawaiian/Japanese overlay. But it's not dying, just frozen in time. To the extent Hilo has a "tourist area," this is it. The biggest (and most "local style") farmers market on the island is here, and is a must see for fresh tropical fruit and flowers. Some other interesting attractions include the Lyman museum, Tsunami museum, Imiloa astronomy museum, Hilo Zoo. Cruise ships come in every now and then, too. Great family-style local style and Japanese restaurants all over town. If you want to experience "old Hawaii", this is a fun way to spend the day.
I say "the day" because there's not a single 3.5+ star hotel anywhere in Hilo (not counting B&Bs), so it's strictly a day trip from the Kona side for most visitors. This is not a bad thing. In in fact, it probably contributes to keeping "old Hilo" special.
The weather here tends to be wet and humid - very green and tropical.
Hilo - south / eastern / inland side. This region includes the area from the airport, out to the eastern edge of town This is the newer part of Hilo. It's clean, modern, and feels like any suburban city in America (again, with a heavy Hawaiian/Asian overlay). Good place to shop at the mall, grab an ice cream, and catch a movie. But that's about it.
A little less humid here (it's farther from the water) but still tropical.
Kona - south or "downtown" (aka Kailua-Kona). This is the main tourist trap on the Big Island. It's kind of stuck in the 70s, infrastructure-wise and culturally. Not much to do here, but the Hulihee Palace is interesting to see how Hawaiian royalty lived, and the Marriott King Kamehameha hotel has some nice Hawaiian art and artifacts inside. Most of the restaurants here are mediocre.
The weather is hot and humid here, but not very rainy.
Kona - far north (aka Kohala coast). This isn't really Kona (which as stated above really means the town of Kailua-Kona), but most people refer to this as "Kona side" anyway. This is the "upscale visitor" part of the island. Think of manicured resorts (Four Seasons, Ritz Carlton, Hilton, Marriott, Mauna Kea) with fancy restaurants, pools and golf courses. Check out A-Bay for ancient Hawaiian fish ponds and a cute little salt-and-pepper beach.
Except for the man-made, tropical-looking resorts, everything is lava, with brown grass growing out of the rocks. It feels a bit like Arizona except it's Hawaii.
The weather is hot and dry here.
Short answer; Hilo wet, Kona dry. There are many more comparisons such as Hilo more industry, Kona not so much. Overall Kona my most favorite place of all the Hawaiian islands; quiet, gorgeous ocean views, uncrowded, Hilo lovely, uncrowded, old Hawaii as possible. Best answer, go ASAP; you'll not regret it.
I don't know Hawai‘i island all that well, and defer to previous answers for that. But one big difference now is coqui frogs. Accidentally introduced in the '80s, the climate on the Hilo side has allowed them to proliferate. It has gotten so bad in some areas that hotels provide ear plugs to their guests.