sn and dm stands for the company who makes the chip ,
https://en.m.wikipedia.org/wiki/7400_series. dm7416n is good to use

SN is typically Texas Instruments and DM is National. I have seen National use the SN prefix on occasion but its almost always TI.
The N on the end means dip package. Yes, those parts will interchange.

Eons ago and continuing to today, lots of corporate parts buyers (including ours) bought IC's using only the exact part number and are/were not allowed to substitute parts. To get the easiest and most common parts - we always called out the Texas Instruments parts such as an SN7416N. Many of the parts manufacturers knew this and some started multi-part number marking them to catch their own part number along TI part numbers (e.g. National). Even Motorola did it sometimes and we would get 7400 series parts with Texas Instrument's SN prefix and Motorola logo's on them.

Do you know why Motorola throws the '1' suffix at the end of their parts in the 70s and 80s? This tends to confuse people looking for a replacement 741541, 45261, or whatever?

Not a clue.
Probably for the same reason they put a "1" in their prefix for 4000 series parts. So their 14049 was the same as everybody else's 4049.... I think the reason was "because they can".