Wise cracks aside, lemme know if I understand correctly: a modem and a
router are interchangable.
No.
Technically a modem is a dumb device that
only accepts and forwards connections,
No.
The term "modem" is short for "modulator-demodulator." Technically,
it's a device that converts the analog signal on the telephone line to
the digital signal needed by a computer, and vice-versa. Technically,
any device that doesn't do that analog to digital conversion is not a
modem (but see below).
a router is a computer itself
that manages the connections/packets more intelligently. I believe my
modem, is a genuine router. Let me know how I can verify it.
A router is a networking device. It's one type of device that permits
you to send data from one computer on your network to another computer
on the network. Note that a router has nothing to do with connecting
to the phone line (or cable service). As a matter of fact, by itself,
it has nothing to do with the internet at all.
A device that connects to a high-speed internet connection is properly
called a "gateway," not a modem, because that high-speed internet
connection is digital to begin with. So there's no analog to digital
conversion, no modulating or demodulating is required, and the term
"modem" is technically inappropriate.
However, the difference between a modem and a gateway is not widely
known, and the term "modem" is widely used for both types of devices.
Some people strenuously object to this usage, because it's not
technically correct. My personal feeling is that, leaving aside the
analog to digital conversion issue, both devices do essentially the
same thing--they connect a computer (or network) to the internet.
Since there is no term that is really correct for any device that
connects a computer to the internet, and since the term "modem" is so
widely used for this, I think insisting that a gateway not be
called a modem is just rigid and inflexible. Despite the original
meaning of the term, for all practical purposes, calling that DSL or
cable device on your desk a "modem" is far and away the best thing to
do. Like so many English words, the word "modem" has changed its
meaning over time.
A router by itself doesn't connect to the internet. If you have a
device that does, it's a combination of a router and a DSL modem in a
single box.
Such combination boxes are becoming more common, but some of us have
individual boxes. In my home, I have a separate cable modem and a
router.