While people who reject the gospel of Jesus Christ sometimes claim jeeringly but mindlessly that the gospels are inconsistent with each other, people who accept the gospel sometimes read into the apparent discrepancy in word usage unsubstantiated extra-biblical meanings. Let's carefully examine the synonymity and equivalence of 天国 and 神国 by comparing the relevant parallel sayings of Jesus as recorded in the synoptic gospels (Matthews, Mark, Luke). We find that Matthew's gospel tends to use Kingdom of Heavens (but also Kingdom of God) and Mark and Luke exclusively use Kingdom of God. They are synonymous words describing the reigning of God over all levels of heavens.

(1) Compare

Matthew 4:17 从那时候耶稣就传起道来、说、天国近了、你们应当悔改。

with

Mark 1:14 约翰下监以后、耶稣来到加利利、宣传　神的福音、
1:15 说、日期满了、　神的国近了．你们当悔改、信福音。

(2) Compare

Matthew 4:23 耶稣走遍加利利、在各会堂里教训人、传天国的福音、医治百姓各样的病症。
Matthew 9:35 耶稣走遍各城各乡、在会堂里教训人、宣讲天国的福音、又医治各样的病症。

with

Luke 8:1 过了不多日、耶稣周游各城各乡传道、宣讲　神国的福音．和他同去的有十二个门徒、

(3) Compare

Matthew 11:11 我实在告诉你们、凡妇人所生的、没有一个兴起来大过施洗约翰的．然而天国里最小的、比他还大。

with

Luke 7:28 我告诉你们、凡妇人所生的、没有一个大过约翰的．然而　神国里最小的比他还大。

(4) Compare

Matthew 11:12 从施洗约翰的时候到如今、天国是努力进入的、努力的人就得着了。

with

Luke 16:16 律法和先知、到约翰为止．从此　神国的福音传开了、人人努力要进去。

(5) Compare

Matthew 13:11 耶稣回答说、因为天国的奥秘、只叫你们知道、不叫他们知道。

with

Mark 4:11 耶稣对他们说、　神国的奥秘、只叫你们知道、若是对外人讲、凡事就用比喻．

and

Luke 8:10 他说、　神国的奥秘、只叫你们知道．至于别人、就用比喻、叫他们看也看不见、听也听不明。

(6) Compare

Matthew 18:1 当时、门徒进前来、问耶稣说、天国里谁是最大的。
Matthew 18:3 说、我实在告诉你们、你们若不回转、变成小孩子的样式、断不得进天国。
Matthew 18:4 所以凡自己谦卑像这小孩子的、他在天国里就是最大的。

with

Luke 18:16 耶稣却叫他们来、说、让小孩子到我这里来、不要禁止他们．因为在　神国的、正是这样的人。
Luke 18:17 我实在告诉你们、凡要承受　神国的、若不像小孩子、断不能进去。

(7) Matthew who usually used Kingdom of Heaven also used Kingdom of God interchangeably in the same context:

See
Matthew 19:23 耶稣对门徒说、我实在告诉你们、财主进***天国***是难的。
Matthew 19:24 我又告诉你们、骆驼穿过针的眼、比财主进　***神的国***还容易呢。

and compare it with

Mark 10:23 耶稣周围一看、对门徒说、有钱财的人进　神的国是何等的难哪。
10:24 门徒希奇他的话。耶稣又对他们说、小子、倚靠钱财的人进　神的国、是何等的难哪。
10:25 骆驼穿过针的眼、比财主进　神的国、还容易呢。

See Kingdom of God also in Matthews:

Matthew 12:28 我若靠着　神的灵赶鬼、这就是　神的国临到你们了。
Matthew 19:24 我又告诉你们、骆驼穿过针的眼、比财主进　神的国还容易呢。
Matthew 21:31 你们想这两个儿子、是那一个遵行父命呢。他们说、大儿子、耶稣说、我实在告诉你们、税吏和娼妓、倒比你们先进　神的国。
Matthew 21:43 所以我告诉你们。　神的国、必从你们夺去．赐给那能结果子的百姓。

Lastly, described below is a side note about the translation of Kingdom of Heaven instead of Kingdom of heavens.

In the Hebrew tradition, there are three levels of heavens: the atmospheric heaven (what we call sky), the celestial heaven (what we call cosmos), the third heaven (what we call 天国 for the Kingdom perspective or 天堂 for the locale perspective). Thus Genesis opens with the declaration: in the beginning, God created heavenS and earth. We interpret the plural heavenS as the first two levels as noted above.

In the NT time, the Koine Greek used to write the NT books described the Kingdom of Heaven(s) as batheleio (kingdom) ouranoun (of heavens). Let's not forget the Hebrew mentality of the word when Matthew penned down Kingdom of heavens, which is customarily translated in English as Kingdom of Heaven (notice the capital H). Whether heavens or Heaven, both connote the same meaning that God is reigning over all levels of heavens.