"fast" computer connected under the same user account to the ISP or Internet, using he same phone line, and at nearly the same time ?

What about Modem settings ?

Same security settings ? Same browser ?

Try the "tracert" test to the same target on both computers and compare the results.

Usual reasons for slow connections, affected by various degree:
* Incorrect IP/DNS settings.
* Inappropriate security settings.
* Phone line noise.
* Viruses and parasites infection.
* Too many "not required" startup programs and processes, "scheduled tasks", etc.
* Choice of browsers and/or versions.
* Busy servers at the other end, and at your ISP.
* Your processor speed.