How has biking changed in the past 15 years?
A few differences off the top of my (mtb) head:

Materials:


Hydroforming of aluminium tubes
Wider use of carbon for frames and wheels, not just top end bikes and on aggressive mountain bikes


Technology:


Disc brakes, both hydraulic and mechanical disc brakes are standard
on mountain and hybrid bikes but slowly gaining momentum on touring,
road and cyclocross bikes
Electronic shifting on road bikes, Shimano D12 and other brands use servos to power shifting for more responsive and lighter changes
Clutch derailleurs on mountain bikes to eliminate chain slap
Dual chamber air sprung suspension, dual chambers provide negative pressure
Remote everything, cockpits are now designed so that all manner of adjustments can be made to a bike on the fly such as seat height and suspension firmness
Dropper posts on mountain bikes, while gravity posts have been around for years hydraulic posts are now common for dropping and raising saddles on the fly.
Thru axles are slowing taking over QRs as standard on mountain bikes and have less flex than QRs
Tubeless wheels, no longer do many wheels and tyre sets require tubes
Various different full suspension designs, but there's no best design
Tapered head tube on mountain bikes to provide more stiffness
Narrow-wide chain rings to support 1x drive trains
Press fit headsets and bottom brackets, while headsets are a bit older and been available 15 years ago there are now numerous standards of press fit bottom brackets
Through axle crankshafts (what's the right wording here?) BBs no longer have an integrated spindle and that is part of the crankset, again for stiffness


Geometry:


29" mountain bikes to roll over things better
27.5" (650B) mountain bikes to roll over things a little less better but to turn better
Slacker head tubes on mountain bikes for more stability downhill


Bike Types:


Cyclocross European mud racing on road bikes has become popular across the world and now most brands carry a cyclocross off road bike in their range, good on both road and gravel and hardy commuters
Enduro European gravity racing paired with climbing (either aggressive XC or mostly downhill racing however you want to look at it) has become a term to describe previous all-mountain bikes. Usually fluro colours with about 160mm of travel
Fat bikes large tired (wide) mountain bikes


Performing Enhancing Drugs


Apparently people no longer take them, it's no longer the in thing to do


The Olympics:


BMX became an Olympic sport


Safety


All mountain/endure helmets providing back of head protection