A terrific subject indeed.
Understanding the "kingdoms" is a
must in understanding the theme of the Bible. I recommend studying their
differences first, and as distinctions are made, then concentrate on their
similarities. (You can come back to that later.)
Don't be fooled by articles criticizing scholars. The scholars NEVER have been a final authority about anything. However, do watch out for misinterpretation (or misrepresentation) of gospel passages (there are several above).
When a man (oh, say David Reagan) doesn't have a good handle on the scriptures, he tends to misrepresent things that he thinks he understands. When he failed to note
WHY Matthew is the only book containing the phrase
kingdom of heaven, he headed down the wrong road (which, you know, always leads out at the wrong place

).
A good place to
start:
1. God and heaven are not the same. They have never been the same, nor will they be. God is a spirit. Heaven is a place. This distrinction can just as simply be made in the kingdoms.
Now notice that in Jesus Christ was two natures - two different natures. Compare this with the two natures of the kingdoms and you'll be able to explain what Reagan couldn't in his paragraph on parallel passages.
Simply put, both kingdoms (and natures), physical and spiritual, were present in the Lord Jesus Christ while He walked the earth. They will also be present in the future when He is back. (Study this out until you have it, then move on.)
2. Why is Matthew always talking about a kingdom? The phrase the kingdom of heaven is connected with the preaching of Christ and His apostles. Start digging there.