Presupposing a difference between the two was popular towards the beginning of the 20th century. Most modern scholars, however, conclude there is no difference. A simple parallel study of the New Testament usage shows why.
- Matthew was written to the Hebrews and refers mainly to to the 'Kingdom of Heaven' (with the exceptions of Matthew 6:33; 12:28; 19:14,24; 21:31,43).
- The rest of the New Testament only ever uses the term 'Kingdom of God'.
- When the Gospel accounts are viewed in Parallel, in the places where Matthew uses the term 'Kingdom of Heaven', the Luke uses the 'Kingdom of God'.
As Jesus likely spoke in a language other than the Greek the writers of the Gospels used, the Holy Spirit inspired each writer to translate whatever language Jesus used into these respective phrases. Hence, there is also no contradiction, because the exact language Jesus spoke is not represented.
See also the answers to the same question at BH.SE.
George E Ladd was a prominent theologian who wrote on the topic of the Kingdom of God. In his notable work, "The Presence of the Kingdom", he laid out what many have come to call the 'already but not yet' approach to understanding the concept of the Kingdom of God.
In a footnote #11 in chapter 4, he writes,
No difference of meaning is to be seen between "kingdom of God" and "kingdom of heaven" (Greek: the kingdomofthe heavens) although the latter may place somewhat more emphasis on the trancendental source and character of the Kingdom. It is the Kingdom which comes from heaven and enters this world (H. D. Wendland, Eschatologie , p. 15). The difference of expression is linguistic, reflectingthe Semitic and Greek elements in the Gospel tradition. "The Kingdom of the heavens" a is Semitic idiom which would be meaningless to the Greek ear. Matthew alone has the Semitic idiom (34 times) except for some manuscript tradition in John 3:4. "Kingdom of God" is found everywhere in Mark and Luke as well as in Matthew 12:28, 19:24 (?); 21:31, 43. Probably Jesus favored the semitic form of the expression, thus following the usual rabbinic form.
GE Ladd, The Presence of the Future, Revised Edition, p 110.