The technology could go absolutely anywhere. Lists are usually off-topic, so I won't just suggest a list of possible inventions - I'm sure you can imagine them anyway. Instead, let's look at how the world might be a little different, and how that might direct the development of science and technology.
If you intend to go with a very large deviation between inertial and gravitational mass then probably your biggest problem would be earthquakes. Lots and lots of earthquakes. The reason for this is that different elements on Earth will attempt to follow different orbits around the Sun. Normally, if you combine your two equations the two $m_{2}$ terms cancel and you get an orbit that doesn't depend on the nature of the satellite. Unfortunately in your case, you have $m_{g2}$ and $m_{i2}$, which do not cancel. At the extreme end, if the largest values of this ratio differ from the smallest by a factor of thousands then the surface of the Earth will be violently ripped off into space. (At the surface of the Earth, the gravitational force due to Earth is about 1600-1700 times the gravitational force due to the Sun.) For values less than this, but still large, the surface of the Earth will only almost be ripped off into space - hence, earthquakes. Gravitational "stirring" of the Earth's core and mantle will also be exaggerated by the same ratio, increasing the heating effect due to friction. For a large enough ratio, this frictional heat generation might rival radiogenic heating (which is dominant on Earth), again leading to earthquakes as it the convection of this heat that leads to plate movements. 921
This means that as tempting as it might be to build giant, low-tech skyscrapers out of "gravitationally light" materials, it probably isn't all that safe.
I would also expect a scientific lull leading to a delay in post-Newtonian science and technology, as deriving the equivalent of <PERSON>'s second law might be a lot harder. Attempting to derive the laws of motion by experiment will yield different results for different test materials and it would probably take some time to isolate the factor responsible (probably wasting a LOT of time on shape). We only really understand the concept of inertial mass in reference to this law, so plucking the idea from thin air in a world in which inertial mass cannot be related to any other known property (such as weight in our own) would be a truly insightful leap of genius. In your setting you can solve this issue by just having everything start a little earlier and have historical societies a little more advanced than us up to <PERSON> and then level after.
Flight will probably be the single most significant technological deviation from our own history - with the right materials even some of <PERSON> designs might have achieved flight. This might have profound consequences for society. The relative ease of travel afforded by flight has been credited with bringing the world together in the second half of the 20th century and for reducing nationalist and xenophobic sentiment. Your world may be a more tolerant place, if not necessarily more peaceful. Contrary to the previous paragraph, this might actually lead to faster advances in scientific development as scientists are more inclined to collaborate across borders.
For $m_{g}/m_{i}$ values that don't destroy the earth, spaceflight is probably still off the table until the development of liquid fuel rockets - Napoleonic Wars through American Civil War era solid fuelled rockets are wildly imprecise by the standards needed for spaceflight. It might be possible for a highly reactive, extremely "gravitationally light" metal (I'd go for aluminium) to always be found in nature bonded to a "gravitationally heavy" element, such that the compound isn't immediately stripped off into space. Such a strongly bonded ore could only be refined by electrolysis making it very rare in your setting, but not unheard of. By the time of your story enough material might have been gathered for the world's first attempt at a steam powered rocket launch - it would make a nice news item even if you didn't want your players going into space.
That's probably a long enough post by now. Try to keep the mass ratios within a reasonable limit (10s or at most 100s) and just use your imagination. 693