On a Kindle where JBPatch still works (we are talking firmwares from around 2013) you go with .mobi - because of JBpatch's hyphenation (best in class for a Kindle) and page formating options.
On every other firmware version you go with azw3 - because you can add hyphenation (soft-hyphens) to it using a Calibre Plugin called "hyphenate this".
As for the "why?" that almost lies embedded in this two answers -
Amazon uses three (? new one for .kfx) entirely separate eBook Viewers (render engines) on Kindles.
.mobi is the best possible (objectively, looking at the featureset) format for one.
.azw3 is the best possible (objectively) format for another one
and
.kfx is so closed down - lets call it "in a class of its own".
When ixtab (the author of JBPatch) worked on making tweaks possible to the viewer engine for .mobi - Amazon reacted highly agressively and confrontational, changing the obfuscation of the code repeatedly, in short term intervals.
(Think of it as "not only was the code they use scrambled", but they took the effort to scramble it differently each time ixtab made some advances.)
Nowadays we know why - because Amazon wants to declare some formating features "Amazon exclusive" and wont allow the best eBook viewer to be used with formats that arent distributed by them exclusively.
Its called company politics.
(The jailbreak scene is allowed custom screensaver pictures and thats pretty much it..
(There are other uses for a jailbroken Kindle (VNC, some dedicated Apps) but those arent in wide circulation.) - Also, from time to time Amazon allows us to educate others how to use some breadcrumb features they leave accessible for some time, before they randomly close them down (such as actually useable font scaling or embedding user picked fonts).)
Take this as an opinion piece..
edit: Also - you can use third party viewers on jailbroken Kindles which allow you to use all open and reverse engineered formats (epub, azw3, ...) - but you loose ease of use, dictionaries, Amazons annotation feature (which isnt half bad...), so theres that. But at least once a Kindle is jailbroken, you know that you'll always have this as a fallback.
edit2: As for the larger azw3 files. Amazon promoted .azw3 as a container for both .mobi and .kf8 for its entire format life. When you think of .azw3 the way current Kindles see them, you are actually thinking of .kf8 in the .azw3 wrapper. .azw3 mostly becomes that much "bigger", because there are both the .mobi and the .kf8 version of a book embedded in one .azw3 file.
Amazon used it as a transition technology (to get away from .mobi and .prc which had a legacy of being "rather open").
Calibre has an option to only package the .kf8 file in there (actually it is its default .azw3 setting for some time now), which almost halfs its filesize.