Hamd literally means praise, the opposite of condemnation and dishonor. 
Praise is a sort of action, not just an utterance of words. We praise an object, person, action or event that we approve of and are satisfied about. Therefore, praise is the result of satisfaction which makes it an existential act or phenomenon. When we are satisfied of somebody we adore them. This love is what inclines us to a set of actions, sayings, qualities and virtues identical to those of the beloved which finally results in absolute servitude, respect and taslim (submission) to that entity; this is when hamd has become complete and reached its final level.
The more knowledge (ma’rifah) one gains the more he realizes that all praises, except for God, are futile, for the good in everything originates from Allah; therefore, God is worthy of the greatest praises and inevitably all praises lead up to Him as the ultimate source of good.
Tasbih is characterized as absolving God and purifying Him of all deficiencies, needs, parallels and all qualities that don’t conform to His holiness and the highest tasbih is to void God from all qualities as praising God through His qualities is a form of shirk. Hence, tasbih means to move from shirk to tawhid and from His qualities to His essence (dhat). 
Essentially, subhan is a noun like ghufran and is sometimes used as one of Allah’s names (qualities) similar to how subbuh and quddus are names of Allah.
And in the phrase subhan’Allah, the word subhan is mafuul mutlaq or the substitute of a verb (Arabic syntax), so the definition of subhan’Allah is actually “sabbahtullaha tasbihan”, meaning, I purify God, a complete purification, which is to absolve God of what doesn’t conform to His holiness.
In simple terms, subhan’Allah means: I absolve God of what doesn’t conform to His holiness.
And so based on the above, hamd means praising God and tasbih is to purify and absolve – void - Him. These two concepts can be frequently seen together in most ayahs of the Qur'an, dhikrs of salat, supplications, etc., because they are complementary to each other.
In addition to the fact that these dhikrs have been mentioned numerous times in the daily rituals, they are also recommended (mustahabb) and are considered to be two of the main and fundamental dhikrs in Islam. In a hadith regarding the special status of hamd and tasbih, Imam Ali (A.S) states: “التسبيح نصف الميزان و الحمد لله يملأ الميزان”; tasbih is half of the scale and hamd fills it. 
In closing, we must note that pondering about the meaning of dhikr, intuitive remembrance of God and applying its meaning in our mind and soul is the most important thing intended of dhikr and the best dhikr is dhikr in action. We also know that dhikr is more than utterance of words (wird) and that its great benefits are rooted in the true manifestation of its reality in life as opposed to merely repeating it without paying attention to the meaning.
 Lisan al-Arab, vol. 3, pg. 155, Jame’ al-Ahadith Software.
 For further explanation, see Question 9803 (website: 9773).
 Musavi Hamedani, translation of Al-Mizan, vol. 13, pg. 5, The office of Entesharate Eslami Publications, Qom, 1374.
 Kuleini, Al-Kafi, vol. 2, pg. 506, Dar al-Kutub al-Islamiyyah, Tehran, 1365.