Do they use vastly different algorithms? Are any of those tool particularly ideal for one type of scenario over another?
In: 2
The differences are more historical than technical.
ZIP is the oldest of the 3.
What gave it supremacy in the 1990s was the original software shipped with a full specification for it.
That meant that other software developers could write their own zipping/unzipping software which would be compatible with the original PKZIP.
That led to the .zip file being supported ubiquitously, on basically every type of computer around.
That was a real rarity back in the day: it was very rare for a file format to allow for real, easy, painless interoperability with different computer systems.
Due to its age, ZIP originally supported only a few (now considered) old compression systems, which are not so great by today’s standards.
Newer and more advanced compression systems have been added into the file format over the years, but not universally.
Some ZIP software doesn’t support all of the newer stuff, which means that a lot of ZIP software will default to using the older compression systems, to get maximum interoperability.
RAR was a competitor to ZIP in the 1990s, trying out different compression systems to get an edge on ZIP.
Unlike ZIP, RAR is a proprietary format, and never achieved very wide support because of that.
In the late 1990s and early 2000s, RAR was legitimately superior to ZIP in its compression abilities, though it still never took a strong strangehold (outside of certain countries and use cases like piracy) because it didn’t have wide support or work out-of-the-box on people’s computers.
These days, ZIP and RAR support more or less the same compression systems.
7z is only about 20 years old.
It was a format developed specifically for a new compression system being researched called LZMA, which beat the pants off of all the older compression systems.
Like ZIP, it was fully specified (and the source code was even released into the open).
Due to the open source nature of 7z, the LZMA code was incorporated into a lot of other file formats, like ZIP and RAR.
Because 7z had an open specification like ZIP, it immediately reached wide support on all computing platforms.
However, by the time 7z came about, operating systems were already including ZIP capabilities directly baked into the operating system (there was no longer any need to download compression software).
To use 7z, you needed to installed the 7z software.
To use ZIP, you didn’t need to install anything: it was already part of your operating system.
Because people are very fond of convenience, 7z would never get as popular as ZIP, even if it was superior.
And now, it’s possible to use LZMA with ZIP, so….
Just use ZIP.
Other differences are not very interesting.
All 3 bundle multiple files into one archive.
All 3 allow splitting large archives into multiple pieces.
All 3 allow the use of the same best compression schemes.
All 3 allow secure password-based encryption.
There are very few interesting technical differences between them.
(Edit: there are a few minor differences in supported compression schemes.
For example, ZIP and RAR have compression schemes specific to compressing .wav files, whereas 7z doesn’t.
These very specific use cases very rarely come up these days.
The best general-purpose compression schemes, DEFLATE, LZW, bzip2, PPMd, LZMA, that will be used in 99.9% of cases, have equal support among all 3 formats)
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