There are many differences between al-hamd and ash-shukr, however that which makes them exact is that al-hamd is praising with the tongue, and praising everything that is jameel (beautiful), and as for ash-shukr then it occurs with both the tongue and action.
Therefore, it is not said that so and so praised Allaah, the Mighty and Majestic, with his action. It is necessary that al-hamd occurs with the tongue. However, ash-shukr is possible with the tongue and it is possible with action. He, the Mighty and Majestic, said, "That you be grateful to Me and to your parents" (Luqmaan 31:14), and He also said, "Work, O family of Daawud, (out of) gratefulness" (Saba 34:13).
From the point of view of where it occurs, ash-shukr is more general than al-hamd, because it includes the praise of the tongue and the action, and al-hamd is more specific because it only occurs by the tongue.
And as for that which is praised, (then from this angle) al-hamd is more general. So this (matter) is from the issues about which the scholars have said that between the two there are generalities and specifics, that they are both in agreement in one thing, but separate from each other in another thing.