Is the only difference between「と言います」and「と申します」politeness?

Question

I've learned several forms of introduction, from the simple "茂{しげる}です" ("I'm Shigeru") to "私{わたし}の名前{なまえ}は茂です" ("My name is Shigeru"), to "茂と言{い}います" ("I am called Shigeru").

Now I have come across "茂と申{もう}します", which seems to be more or less equivalent in meaning to と言います, differing only in politeness and formality. However, I could not find a definitive explanation as to how exactly they differ, and when one should be preferred over the other. This answer on HiNative states:

"... to iimasu" is used in formal situations.

"... to moushimasu" would be a polite way of saying it.

This answer on YesJapan states that "... to moushimasu" is more polite than "... to iimasu" and that it is thus preferable.

These two seem a bit contradictory, as the way it is phrased in the first link suggests that "to iimasu" is more polite, while the second suggests "to moushimasu" is more polite.

Thus my question: Are these two expressions equivalent in meaning and only differ in "politeness"? If so, which one is more polite and when should one be preferred over the other?

Answer 1

Simply put, と申します is politer than と言います. In business settings, it should be safer to use the former whereas the latter may sound natural e.g. when you introduce yourself at a casual party hosted by your friend.

The difference is 丁寧語（と言います） and 謙譲語/丁重語（と申します）. Quoting a website,

丁寧語の代表は「です」「ます」です。 「です」「ます」を使うことで、 聞き手やその場の状況に配慮していることを表します。

丁重語（謙譲語Ⅱ）とは 非常に改まった場で用いる敬語です。 例えば 「よろしくお願いします」は 丁寧な言い方で、よく使われます。

それに比べ 丁重語を使った 「よろしくお願いいたします。」は 改まった場で使う言い方となります。

「改まった場で使う」とは 入学式や卒業式、結婚式などでの スピーチを思い浮かべると わかりやすいと思います。

丁重語（謙譲語Ⅱ）は フォーマルな服装をしなければならない場で 使われます。 また、 大勢の人の前で話す時にも使われるので、 大切な敬語となります。

So both are polite forms 'indicating considerations to the circumstances', but 丁重語 is a form used in more formal settings (giving a presentation, formal speech, etc.).