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 天國 and 神國 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 從那時候耶穌就傳起道來、說、天國近了、你們應當悔改。

with

Mark 1:14 約翰下監以後、耶穌來到加利利、宣傳　神的福音、
1:15 說、日期滿了、　神的國近了．你們當悔改、信福音。

(2) Compare

Matthew 4:23 耶穌走遍加利利、在各會堂里教訓人、傳天國的福音、醫治百姓各樣的病症。
Matthew 9:35 耶穌走遍各城各鄉、在會堂里教訓人、宣講天國的福音、又醫治各樣的病症。

with

Luke 8:1 過了不多日、耶穌週遊各城各鄉傳道、宣講　神國的福音．和他同去的有十二個門徒、

(3) Compare

Matthew 11:11 我實在告訴你們、凡婦人所生的、沒有一個興起來大過施洗約翰的．然而天國里最小的、比他還大。

with

Luke 7:28 我告訴你們、凡婦人所生的、沒有一個大過約翰的．然而　神國里最小的比他還大。

(4) Compare

Matthew 11:12 從施洗約翰的時候到如今、天國是努力進入的、努力的人就得着了。

with

Luke 16:16 律法和先知、到約翰為止．從此　神國的福音傳開了、人人努力要進去。

(5) Compare

Matthew 13:11 耶穌回答說、因為天國的奧秘、只叫你們知道、不叫他們知道。

with

Mark 4:11 耶穌對他們說、　神國的奧秘、只叫你們知道、若是對外人講、凡事就用比喻．

and

Luke 8:10 他說、　神國的奧秘、只叫你們知道．至於別人、就用比喻、叫他們看也看不見、聽也聽不明。

(6) Compare

Matthew 18:1 當時、門徒進前來、問耶穌說、天國里誰是最大的。
Matthew 18:3 說、我實在告訴你們、你們若不迴轉、變成小孩子的樣式、斷不得進天國。
Matthew 18:4 所以凡自己謙卑像這小孩子的、他在天國里就是最大的。

with

Luke 18:16 耶穌卻叫他們來、說、讓小孩子到我這裡來、不要禁止他們．因為在　神國的、正是這樣的人。
Luke 18:17 我實在告訴你們、凡要承受　神國的、若不像小孩子、斷不能進去。

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

See
Matthew 19:23 耶穌對門徒說、我實在告訴你們、財主進***天國***是難的。
Matthew 19:24 我又告訴你們、駱駝穿過針的眼、比財主進　***神的國***還容易呢。

and compare it with

Mark 10:23 耶穌周圍一看、對門徒說、有錢財的人進　神的國是何等的難哪。
10:24 門徒希奇他的話。耶穌又對他們說、小子、倚靠錢財的人進　神的國、是何等的難哪。
10:25 駱駝穿過針的眼、比財主進　神的國、還容易呢。

See Kingdom of God also in Matthews:

Matthew 12:28 我若靠着　神的靈趕鬼、這就是　神的國臨到你們了。
Matthew 19:24 我又告訴你們、駱駝穿過針的眼、比財主進　神的國還容易呢。
Matthew 21:31 你們想這兩個兒子、是那一個遵行父命呢。他們說、大兒子、耶穌說、我實在告訴你們、稅吏和娼妓、倒比你們先進　神的國。
Matthew 21:43 所以我告訴你們。　神的國、必從你們奪去．賜給那能結果子的百姓。

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 天國 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.