At the time I also remember hearing about how the internet was just chat rooms for child molesters to find victims. Pretty similar to the way people say Bitcoin is just for criminals and drug users now.
It wasn't until WorldWideWeb was released that the masses started seeing the internet had some real potential. Once netscape was released and it was simple for anyone to hop on MTV.com and watch Beavis and Butthead videos, it spread like wildfire. Things like ICQ started popping up to make conversations online easier with friends (I still remember how amazing it was the first time I could see when my friends were online), and everybody started slapping together their own websites hosted by free services like Geocities. It took a long time for what now seems obvious, to become a reality.
What innovation will get Bitcoin to the next level? Who knows... Maybe a social network linked to Bitcoin Addresses to verify identities, or an ICQ like addition to the client where you can send encrypted messages through the p2p network and see when your friends are online. Maybe an ApplePay like network where merchants can easily accept Bitcoin alongside USD without even being able to tell the difference. Maybe a partnership with Microsoft so that Xbox Live can incorporate a Bitcoin system that allows users to bet on games between each other. Maybe it will be a country that embraces it for their financial markets to become unregulated and draw in outside investments. Maybe it will be schools that embrace it as a learning tool to teach about developing software. There are so many possibilities and the exciting part is that it is up to you to make it happen.
EDIT: When I think about it, Bitcoin really does strike me as the same type of advancement as the internet and mp3s. This is coming from someone that swore by Apple products in the 80s (don't get me started how many of my friends argued with me about Apple then, only to buy every iPod and iPhone that would come out 25 years later), was using the internet before AOL came out, and started listening to mp3s 10 years before the iPod was invented... Also interesting is that when I discovered mp3s, I told all of my friends about them and only 1 of them actually started using mp3s back then. Since I discovered Bitcoin in 2011 and pushed the idea to every person I know, the only friend I have who actually bought some (at $5) and started using them is the same friend who started using mp3s back in the early 90s. Take from that what you will...
Even "backbone" connections that ISPs had back in the 1990s likely were slow when compared to what speeds are available to residential customers for less then $100 per month today. The speed at which the internet back is measured is likely greater then the speed of the internet backbone in the 90's.
I would say that bitcoin was a byproduct of the internet. It was an advancement that was made possible by the internet