申もうし訳わけありません

Meanings

I'm sorry; I apologize

A formal, polite apology used in business and serious situations. Stronger than すみません.

Used to express deep gratitude, often when someone has gone out of their way to help. Conveys a sense of indebtedness.

すみませんN4sense 1
すみません is a lighter, everyday apology or attention-getter, while 申し訳ありません is reserved for formal, serious apologies in business or polite speech.
ごめんなさいN3sense 1
ごめんなさい is a personal, casual apology used with friends and family, whereas 申し訳ありません is the formal expression expected toward superiors, customers, or in business settings.
申もうし訳わけないN2sense 1
申し訳ない is the plain, less formal adjective form, while 申し訳ありません is its polite conjugated form used in formal speech.
あしからずN1
申し訳ありません is a strong, direct apology that admits fault, while 悪しからず simply asks the listener not to be offended by unwelcome news.
ごめんくださいN2
申し訳ありません is a more formal, weighty apology suited to business settings, whereas ごめんください is rarely used this way in modern speech.

申もうし訳わけございませんpolite 丁寧語sense 1
申し訳ございません uses the more formal ございません instead of ありません, making it an even higher-register version of 申し訳ありません for very formal or customer-facing situations.
ごめんなさいplainN3
申し訳ありません is the very formal, humble counterpart of ごめんなさい, used in business or serious situations rather than everyday ones.

From 申し訳 (もうしわけ, 'excuse') + ありません (polite negative of ある, 'to exist'), literally 'there is no excuse'.