The question depends entirely upon in what sense we understand the terms Israel and church, respectively. If we mean Israel and church absolutely, as in true Israel and the true church, then they are one and the same thing (Gal. 6:16). If we only mean by Israel a political entity in which was found the visible artifacts of the Old Covenant administration, then there is an obvious distinction. The church in itself is no political nation, though it once existed under a covenant through which a political nation was instituted.
To dispel any doubts as to whether or not Israel and the church are ever identified with one another, it cannot be denied that Israel is called a church and even the church of God (cf. LXX, 1 Chron. 28:8). For example, Hebrews 2:12, quoting Psalm 22:22 says, “I will declare Your name to My brethren; In the midst of the church I will sing praise to You.” In the LXX version of Psalm 22:22 (LXX, Ps. 21:23) the same term is used (ekklesia), indicating the author of Hebrews used the LXX rather than the Hebrew text in his quotation.
So, to answer the question, “Are Israel and the Church always distinct?” No, they are not. Israel is the church of God (2 Chron. 30:25; 1 Chron. 28:8). Thus, the question becomes, “Were there two churches, then? One for the Old Covenant and one for the New?” But, to ask this question would be like asking, “Does God have two brides?” Or, “Does Christ have two bodies?” The answer to either is most obviously “no.”
In Hosea 2:1-13, God appears to write off His bride. But then, starting in v. 14, He promises her restoration and purification (v. 19). Finally, He promises she will “know the LORD,” an echo from the promise of the New Covenant seen in Jeremiah 31:31-34, “They will not teach again, each man his neighbor and each man his brother, saying, ‘Know the LORD,’ for they will all know Me…” It is this bride for whom Christ died, and also this bride who is called the church (Eph. 5:25-33; Rev. 19:7; 21:2, 9).
In Hebrews 3:1-6, we are told of one house over which Christ now presides, but before Christ there was Moses, who presided over that same house as a servant. What is this house of God over which Moses first served and then over which Christ now rules as a Son? “I write so that you may know how you ought to conduct yourself in the house of God, which is the church of the living God… (1 Tim. 3:15).” The house of God is the church.
Thus, God has only ever had one true church—spiritual Israel—which has existed under both Old and New Covenants. The difference is not in the kinds of churches, as if there were different churches altogether. The difference is found in the two covenants, Old and New, under which the one church of God has existed since the fall of Adam.