I know they're not very similar cars, but they're the car I bought when I was 18 and the car I bought when I was (am) 25. I figure I've owned the WRX long enough now and making a comparison of the two is actually quite appropriate.
Inside the Tiburon, the leather seats and enclosed all-black cockpit which was illuminated with amber lights and had a very sporty and slightly luxurious feeling to it. The sound system was after-market and very teenage-appropriate, with its fancy deck and noisy subwoofers. However the car was completely useless for more than two people. If anyone over 5'6 sat in the back seat, they couldn't sit up straight. There was actually a sticker on the hatch warning you to not close it if people were in the rear seats.
The WRX has a fairly plain cockpit. The controls are all easily enough accessed. The thermometer and gas-mileage readouts give me a slight reminder that my car isn't actually the old 240sx I drove in high school and the radio lets me control my music from the steering wheel. That's nice, even if everything else about the audio is in dire need of an upgrade. The back seat however is great for friends and the sporty looking seats hold you nicely as you take turns at a faster pace than you should.
The Tiburon had a way-too-high geared 6 speed mated to an under-powered V6 with I/H, lightweight flywheel (the stock one was 25 lbs.. seriously) and a stronger clutch. The car was very quick in 1st and 2nd gears. Shifts were fun, but that was due to the upgrades after a blown clutch at 60,000 miles. On the highway 3rd gear was too low to pass people, so passing was done in 4th gear. 5th and 6th gears were both overdrives, so 4th gear was not a very powerful gear and the car was about on par with my old Ka24 powered 240sx (not very powerful) when it came to passing. Going around corners the car was stable, but not very sporty due to its horribly front heavy design. The suspension was uncomfortably stiff on any roads that weren't made of glass. Oddly enough torque steer was never really an issue. Going up canyons on the freeway, 6th gear was fine. However this is likely because at 80 mph in 6th gear I was at 3500 rpms. Going up smaller canyons 4th gear was usually sufficient and I rarely shifted. Gas mileage around town and on the highway were about the same at roughly 23 mpg, despite Hyundai's estimate of 18/26.
The WRX has turbo lag. This is my first turbocharged car and it was so strange for me at first. The car feels like I'm driving a 2.5 Impreza most of the time, but when I ask it to, it comes alive and is one of the most amazing things I've ever driven. The unrefined throw-you-back feel that I get at 3500 rpms is amazing. However it's unfortunate as going up and down freeway canyons it requires a lot of shifting between 4th and 5th gears to allow the turbo to spool up. Commuting to work in traffic is also a huge pain in the ass because of the amount of shifting I have to do. Taking the WRX around turns is amazing as it feels so stable with its stock suspension and much more balanced than the Hyundai did. Driving over pavement the suspension is stuff, but I've never had a back ache after a long drive. The car also explodes out of turns as long as the RPMs stay at the right spot. Gas mileage is slightly better with about 22 around town and 28 on the highway.
Reliability wise it is hard to compare as the WRX has 25,000 miles and the Tiburon had 140,000 miles. I do have to admit it's quite nice not noticing the AC become significantly warmer as the summer goes on, having to hold the rpms high on hot days when stuck in traffic to stop the car from overheating and worrying about electrical issues that once a month or so caused the fuse powering the radio to blow when the passenger door is opened. The WRX hasn't had a single issue yet and appears to be much better engineered that the 10 year old Hyundai.
I've yet to drive the WRX in the snow, but I live in Utah so I will get plenty of opportunity soon enough. The 3000 lb FWD Hyundai with a 62/38::F/R weight ratio never had much trouble in wet snow, however powder was a different story. In the very occasional rain that we get here the WRX has already shown me that it doesn't mind being on the freeway nearly as much as the Tiburon did.
In summary I miss the Tiburon. It was unrefined and underpowered and had looks that screamed "an 18 year old drives this" but it was actually quite nice inside and very easy to drive despite the unnecessarily extra gear. However I also owned the car for 7 years and knew just about everything I could about it. The WRX is just as unrefined, but certainly not underpowered. It has a mature, but still young, look to it and looks like something I could drive well into my 30's. I think my only major gripe about the car is that the interior seriously looks like something from the mid 1990's. Otherwise, the WRX is definitely an appropriate and needed upgrade over the Tiburon and I'm happy with my purchase.