I coincidentally bought an aluminum road bike (Cannondale) about the same time as you got your Giant and with very similar specs. Mine was 3x8 rather than 3x7, but same size chainrings and similar gear range. I rode it until the frame was cracked in an unfortunate incident and replaced it with a relatively modern Felt aluminum bike with 2x10 gearing with a compact double chainring. I actually still use the 3x8 drivetrain components on a commuter bike.
On the plus side, by 1998, many of the really substantive changes in road bike technology had already happened. Combined brake lever/shifters were standard, as well as dual pivot brake calipers, which have a lot better stopping power than some of the earlier single-pivot side pull calipers.
So what's changed?
Brakes
Although disc brakes are not yet standard technology for road bikes, the trend is certainly in that direction. Disc brakes are not something that can be retrofit on an old frame, so if you want those, you do need to get a newer bike.
The primary advantage of discs in better stopping power in rain and overall slightly more consistent performance. That said, the dual pivot caliper rim brakes that you have now have great stopping power as long as the pads are kept in good condition. The only reason I can imagine switching to discs is if you ride a lot in a group with other riders who have discs and you ride in the rain. In wet conditions, those other riders might stop more quickly than you can with your rim brakes and cause a crash. Or if you ride a lot on dirt roads, disc brakes have the advantage of not being affected if your rim gets muddy going through a puddle.
Wheels/tires
Another major trend in the last 20 years is a shift towards wider tires with lower air pressures, even among pro racers. Whereas most road bikes were equipped with 23mm wide tires or even narrower in 1999, now 25 and 28 mm are very common and many bikes have even wider tires. The high quality wide tires have no greater rolling resistance than the narrow ones, and the lower pressure makes the ride much smoother.
Because narrow tires were standard in the 90's, many frames did not have clearance for anything larger than 25, so if you wanted to try wider tires, you might be limited. You'd have to measure your frame to check. And if you're happy with the narrow tires you're on now, it doesn't matter.
Frame design
As you noted, a big change since 1999 is the ubiquity of sloping top tubes. Frame construction has also changed a bit, so a comparable aluminum frame today would likely be a bit lighter than your current one and feel a bit less stiff, though road feel is generally much more dependent on tire construction and pressure than on frame materials. But unless there's something you don't like about your current frame, I doubt it's worth the investment to replace it just because the newer ones look different.
Drive Train
Better chainring design (ramps and pins) has enabled double chainrings with large differences in size, and the number of cogs in the rear continues to increase (now 11 is standard on expensive bikes). As a result triple chainrings have largely disappeared from even the lower tiers of expensive road bikes. The main benefit of the doubles over triples is weight, though. There isn't a big difference in performance. And counterintuitively, you may find yourself shifting between chainrings more frequently on the double than you did on the triple because the switch point is in the middle range where you ride most. On the triple, I find that I ride a lot on the middle ring and mainly only use the big when descending and the small when climbing.
If you find your current gearing is a bit high for you, you can easily switch to smaller chainrings on the current setup. I've run 26/39/50 on those same Shimano triple cranks that you have that came with 30/42/52 originally. You should still be able to buy 39 and 50t chainrings with ramps and pins to work together properly, since Shimano sold Tiagra and Sora groupsets with 28/39/50 triples using the same BCD as the 30/42/52.
A downside to the new drivetrains is that the narrow cogs and chains wear much more quickly than the thicker ones in a 7-speed set up (and are more expensive to replace).
New cranks are also lighter due to the switch from square taper spindles to hollow spindles and external bearings. If weight isn't a concern for you, there's really no advantage.
And of course, if you want to switch to a 2x11 drive train, all of the components can be put on your existing frame, so you can upgrade without changing the frame.
Conclusion
If weight is not a concern, the primary benefit to a new bike would be the ability to have disc brakes, which aren't even standard on road bikes now.
You might find that wider lower pressure tires and lower gears make your riding more enjoyable, especially now that you're 20 years older than when you bought the Giant. But if you're comfortable on long rides the way it is, there's no need to change just to follow the trends.
If you want to cut a bit of weight, you can do it by upgrading parts. Perhaps look at nicer wheels or switch to a hollow spindle crank and external bearing BB such as the Tiagra triple from the 3x9 speed era, which would be compatible with all your existing equipment. Just swap on your current chainrings and you're good to go.
