Indeed they are different as they are distinct and separate although “one”. But, before answering “why” they are, we need to know “who” the God referred to by the writers of the O.T and the N.T.
Remember, that the God of the Old Testament is the Creator God, as in “In the beginning Elohim(God) created the heavens and the earth.”(Gen. 1:1).
The Hebrew word “Elohim” translated in English as “God” is a generic name. And, according to Moses, the writer of the Pentateuch, this particular Creator Elohim was specifically named proprietarily as YHVH Elohim, as in “These are the generations of the heavens and of the earth when they were created, in the day that the LORD God (YHVH Elohim) made the earth and the heavens,”(Gen 2:4).
Furthermore, YHVH Elohim introduced himself with a different name, as El Shaddai to Abraham, Isaac and Jacob, but to Moses he was known as YHVH (Ex.6:2-3).
In the Shema doctrine(Deut.6:4; 11:13–17; Num.15:41), YHVH is the one and only Elohim to be worshipped as he is above all “gods” (Ps 95:3; 96:4;97:9;135:5; Ex. 15:11; IChr. 16:25). This means that, while there are many gods (polytheism), and unlike other nations of that time, Israel has to be monotheistic and worship only one God, this specific YHVH ELOHIM.
Fast forward to the N.T, according to the writer John, this Creator God (YHVH ELOHIM) was further named in Greek as “LOGOS” translated in English as “Word”, and further differentiated from the Father who was “THEOS” in Greek. Notice, “In the beginning was the Word(Logos), and the Word was with God (Theos), and the Word was God (Theos). The same was in the beginning with God (Theos). All things were made by him; and without him was not any thing made that was made”(John 1:1-3).
From the preceding, as the word Elohim was generic, so was the word Theos. But, to further specify who the specific Elohim was in the O.T., YHVH ELOHIM was used to refer to the Creator. YHVH ELOHIM in the O.T. was the only God to be worshipped by Israel at that time until the Father was declared in the N.T., as in John 4:21-25.
But, to refer to Jesus or the Father, the generic “Theos” was further delineated as such: “And the Word ( Logos) was made flesh, and dwelt among us, (and we beheld his glory, the glory as of the only begotten of the Father,) full of grace and truth.”
Furthermore, Paul differentiated the Father as Theos and Jesus as Lord, as in Eph 1:3 “Blessed be the God (Theos) and Father of our Lord (kyrios)”Jesus Christ,”
Likewise, Peter differentiated the two, as in “Blessed be the God(Theos) and Father of our Lord (kyrios) Jesus Christ,”(I Pet. 1:3)
This Creator God, YHVH ELOHIM, Logos, incarnated into the man-Jesus, identified as Kyrios Theos (Lord God). As Jesus was the only one who saw God the Father, rightfully he was the only one to introduce and “declare” the Father to us, as in, “No man hath seen God at any time; the only begotten Son, which is in the bosom of the Father, he hath declared him.”(John 1:18). The Father God was not declared before incarnation except as “Ancient of days” (Dan.7:9-14) and even in that, it was a prophetic vision, not an official introduction as Jesus did.
But notice this, while in the O.T., YHVH ELOHIM was the only God to be worshipped, not others, this time, Jesus said in John 4:21-25 “Jesus saith unto her, Woman, believe me, the hour cometh, when ye shall neither in this mountain, nor yet at Jerusalem, worship the Father. Ye worship ye know not what: we know what we worship: for salvation is of the Jews. But the hour cometh, and now is, when the true worshippers shall worship the Father in spirit and in truth: for the Father seeketh such to worship him. God is a Spirit: and they that worship him must worship him in spirit and in truth. The woman saith unto him, I know that Messias cometh, which is called Christ: when he is come, he will tell us all things. Jesus saith unto her, I that speak unto thee am he”. Here , Jesus confirmed he is the Messiah and that true worshippers shall worship the Father.
This corroborates the fact that the God of the Old Testament to be worshipped was YHVH ELOHIM as he was worshipped in Matthew 2:11, 28:9, 17; Luke 24:52; John 9:38, 20:28, being a deity. But now (“the hour comes and now is”), Jesus himself directed and commanded us to worship the one and only Father God. Even Paul wrote that the “one God” is the Father and The Lord is Jesus, as in I Cor. 8:6 “But to us there is but one God, the Father, of whom are all things, and we in him; and one Lord Jesus Christ, by whom are all things, and we by him. Paul said “one God, the Father” to worship, even as Jesus is also God.
On being ONE, Jesus (our Lord God) and God the Father are NOT one person, as we (being many) will be “one” with them, as well, as in: John 17:11, 22-23 “Holy Father, watch over them on account of your own name which you have given me, in order that they may be one just as we are. Also, I have given them the glory that you have given me, in order that they may be one just as we are one. I in union with them and you in union with me, in order that they may be perfected into one, that the world may have the knowledge that you sent me forth and that you loved them just as you loved me.”
As to why YHVH ELOHIM in the O.T. appeared to be very strict with Israel following the laws, it was because his function as a parent/creator was to show from Adam on, that the man to follow all the commandments was the man-Jesus, not anybody else. The Ten Commandments, according to Paul, were “a schoolmaster to bring us to Christ” (Gal.3:24), i.e., for Christ to teach us that he was the “man” to fulfill all the commandments and to die so that in faith we can be justified and reconciled to God the Father. As we ourselves are adamant to our child to follow us in our “dictates” of good behavior, so was YHVH ELOHIM as we were still “spiritual-children”(I Cor. 3:2; I Pet.2:2). But, in the N.T. the role of YHVH ELOHIM was as a Messiah, Passover, and as a man to be meek and obedient unto death (Phil. 2:8-18). Moreover, his role was to “declare”(Jn.1:18) God the Father to us and his love(John 1:1).
In summary, Elohim the creator has many names, viz., El Shaddai, YHVH ELOHIM, LOGOS, MESSIAH, PASSOVER, depending upon his function. Yet, in Heb. 18:3, “He is the same yesterday, today and forever.” The roles of YHVH Elohim (Lord God), were different in the O.T. and the N.T., yet consistent with the plan of salvation. This particular Elohim was the God referred to in the O.T. Whereas, the Father was the God referred to, in general, in the N.T. They are indeed different, yet “one”, as we are all different, yet “one” with our Lord God Jesus and our God the Father.