First: ḥamd is directed only to someone who is alive, whereas madḥ may be offered for both the living and the dead.
Second: ḥamd is uttered only after a prior act of beneficence, whereas madḥ may precede such an act. This distinction is weak, however, for Allah praises Himself.
Third: Some have claimed that madḥ is confined to the tongue, while ḥamd arises from the heart—on the assumption that ḥamd is tantamount to inner satisfaction, as will be explained. Al-ʿImmānī mentioned this, but it is open to critique.
Fourth: The statement, “I praised Zayd (maddḥtu Zaydan),” is not accurate unless praise has already occurred, unlike the phrase “al-ḥamdu lillāh,” which is a declarative sentence that, by its very utterance, brings praise into existence; it therefore combines both a report and the initiation of praise.
Fifth: ḥamd is the act of informing others of the praiseworthy qualities of the one praised while also loving and approving those qualities, whereas madḥ is merely the act of reporting those qualities. This was stated by Abū ʿAbd Allah Ibn al-Qayyim (1).
Others have said that ḥamd is gratitude for blessings. This was the view of Ibn Jarir, al-Mubarrad, and others, and Ibn Jarir transmitted it both as a marfūʿ report traced back to the Prophet ﷺ and as a mawqūf report from Ibn ʿAbbās (2).