I have a Cannondale CAAD9-5. Over an identical distance with identical power output, how much faster would I go with a better bike?
And what are the things that would lead to the greater speed (just lower weight or other factors as well), and which situations (e.g., uphill, flat, downhill) would I see the greatest gains?
That's really hard to answer. Every year, bike makers come out with claims that this year's model is N% more aerodynamic than last year's, etc. Although they don't always give hard specs on CdA.
Also, we don't know what shape your bike is in. A new bike would have a new drivetrain, and your current one might have a lot of wear, which would make it somewhat less efficient. A new one would have whatever tires are stock, but presumably you've gotten new tires since you bought this bike. Maybe those are top-quality racing tires with even lower rolling resistance than the hypothetical new bike, maybe they're bricks.
Also also, a lot of the improvements that go into bikes from year to year can't be measured in a one-hour time trial. For example, there's been an overwhelming shift toward disk brakes, which offer better stopping power in the rain and better modulation in all conditions. There's also been a trend toward wider tires, which reduce rolling resistance and increase comfort. Your current bike probably can't fit the 35-mm tires that many modern road bikes could. A new bike would probably give you another gear or two in back, so you could get a wider gear range with the same ratio spacing.
Weight alone makes surprisingly little difference in performance unless you're on a steep climb, and even then, it needs to be a considerable amount of weight—say, 1 kg—to make a meaningful difference to an amateur cyclist.
So: your current bike isn't bad. If you put top-quality tires on it, a high-quality chain, and a new cassette, and made sure everything was in good tune, a new bike wouldn't offer much advantage in terms of plain speed, but a new bike might offer other quality-of-life improvements that would make it worthwhile.
I maintain the bikes for an elite rider who funds his career by coaching while he finds a team.
The main thing holding you back on a bike is aerodynamics. The faster you go, the more true this becomes. While climbing a steep hill at a crawling pace, the aerodynamic drag is not relevant but when you're thumping along on the flat, it makes a big difference.
The main aerodynamic drag on the bike is the rider. First you get the position right and then the clothing and helmet, as this is where the biggest aerodynamic gains are made. If you already have good clothing, simply reducing the width of the bars (changing from 42cm to 40cm width, for example) makes a measurable difference. You will see alot of pro riders now with the levers angled inwards too to allow for a very narrow riding profile that is still withing the UCI rules.
After that, deeper & wider wheel rims make a big gain to aerodynamics and it's more than just wind tunnel theory, but it's an expensive upgrade.
Any aero gains to the frame and bars are fairly minimal so don't get too excited unless you are buying the complete package.
How much faster any of the above makes you on a new or on your existing (very smart) CAAD9 depends on your power output and current ability. The faster a rider you are, the bigger the benefit from aerodynamics.
I think you need to decide a bit in what conditions you want to be faster and then go for a matching type of new bike. Spoiler: It's probably easier to get an overall faster bike than a superlight climbing machine these days.
Your CAAD9-5 is probably on the realms of 8-9 kgs, so anything lighter will give you an edge in terms of climbing performance, but is it worth the upgrade? Probably not, 1 kg lighter on the bike gives you less than a minute time gain on a 10 kilometer/10% gradient climb - at 200 watts for a 70-kg rider on a bike with 7.5 instead of 8.5 kilograms you need 71 instead of 72 minutes). The effect of weight is even less up to negligible in the flat.
I should add that sub-8-kilograms is probably harder to achieve than it was in the 2000s heydays (with some light but affordable frames and rim brakes all the way), so a 2023 entry-level (or even mid-tier) bike won't be much lighter than yours except you spend some serious money.
Of course, any modern road bike with actual aero properties such as dedicated tube shapes, deeper wheels and internal cable routing will be faster in a straight line. I didn't find any numbers, however, I guess it'll be measurably faster, just because you are more aerodynamic but not to a life-changing extent. You won't hit 35+ instead of 28 kph on your home straight and you won't zoom away on your group rides, but there sure are benefits.
So, I would argue, you shouldn't get a new bike for speed but you may for perceived riding quality or to just treat yourself. Ergonomics and comfort have moved on since the 2000s (trust me, I have a 2004s aluminum bike on 23mm tires), modern 11/12-speed groupsets offer a wider gearing range (which is great for us mortal amateurs), disc brakes provide better braking feel and power and wider tires will give you better riding comfort.
