2001: Culturally closer to 1981 or 2021?

There is absolutely nothing in common with 1981 in a meaningful and substantial way. The only thing is in 2001 cassette  tapes ( were dying but had a small market) laserdisc ( last year of it and niche ) and vhs (still mostly popular) and tape walkmans(few people) were still manufactured in 1981 but in 81 vhs was mostly for rental and expensive, laserdisc was for rich people or video buffs and walkmans were really popular. You could say home taping recording was common in both eras but mix taping was much less common but not completely dead in 2001.

A small amount of baby boomers were still wearing glasses or haircuts from around 1981.

Punk revival was big throughout 94-2005 nothing stands out in 2001 regarding punk this year other than pop-punk being really popular in the mainstream and a bit more commericalized than earlier. The casual underground hardcore punk hairstyles of 1981 outside of mohawks and crazy spiked hair was mainly plain short hair with no fashion style, slightly spiked hair, bleached hair and frosted tips and shaved heads which actually very visible and mainstream in 2001 by pure coincidence or indirect influence from pop punk. Boy bands like boyzone and the Backstreet Boys had an almost similar haircut as punk bands like the Sex Pistols and DOA. Atari retro plug and play remakes were already a thing in 1999.

80s nostagia starts increasing in 2001 but isn’t substantial enough until 04-06.

Desktop Computers were getting popular and more common place in 1981, and pc gaming was starting to become more of a thing, and PCs were seen as the future and in 2001 internet and PCs were seen in an almost similar way but more mandatory and common. 

Also you could go online in something that operates like the internet in 1981 from your home, as every computer had a modem port, where you could connect to BBS and Usenet and yes you could chat and send messages with people all around the world, find hobby discussion  groups, upload text files and download primitive games. No this didn’t start in 1989-1991 or 93-1995. However it was slow, expensive and would operate like dial up and only nerds, a few people and people who worked in areas where they used online networks were using it. In the late 80s to mid 90s casual computer users increased but it was never fully mainstream or accessible by most people until the internet replaced these system, at least mostly, in 94-95.

But yes many BBS and Usenet groups were still active in 2001 possibly even a few from 1981.

Neither