komplimento

No -ment- here, whatever your other languages are telling you. English, French and Spanish all have a live -ment suffix for making nouns out of verbs, so komplimento reads on paper as komplim- plus -ment- plus -o. It is not. Esperanto has no -ment- suffix at all, and kompliment- is one root borrowed whole. Where English uses -ment, Esperanto uses -ado for the action and -aĵo for the thing, so movement is movado and agreement is interkonsento.

This is the monumento class, and it is a class worth naming: momento, dokumento, instrumento, monumento, argumento, elemento, fundamento - every one of them a single indivisible root that happens to end in the same five letters, and not one of them cuttable. Cut anyway and you get komplim-, monu-, docu-, none of which means anything. That is the test.

The seam that is real, and it is further back. The Latin underneath is complere, to fill up, the same verb that gives Esperanto kompleta and English complete and comply. A compliment was originally the fulfilment of a courtesy - the polite thing you were obliged to say, discharged. Spanish still uses cumplido that way. So komplimento and kompleta are genuine relatives, by history rather than by Esperanto word-building, and you cannot get from one to the other with any affix in the language.

The neighbours, by sincerity. Laŭdi is to praise, and it is weightier - you praise work, courage, a book. Komplimenti is to pay a compliment, which is lighter and more social. Flati is to flatter, and it carries the suspicion that you do not mean it. Fari komplimenton is the ordinary phrase, though komplimenti iun is perfectly good, and komplimente is the adverb for saying something in that spirit.

A note on register. Komplimentoj in the plural, at the end of a letter, is the old-fashioned "my regards" - Miajn komplimentojn al via edzino. The letter writing and register article covers where that sits on the formality scale, which is roughly where kind regards sits in English and rather further up than a learner expects.