Israel became a recognized state in the Mandate for Palestine (a British-controlled territory under Mandate since the League of Nations, and prior controlled by the Ottomans) by declaring independence following the expiration of the British Mandate. The story to get to that point is a long one, but the shortest version I can give is that after the British controlled the area for roughly 25 years, they gave up on trying to mediate the conflict (having proposed a single federal state, a two-state solution, etc.). They asked the UN to mediate the conflict between Jews and Arabs. At the time of the UN investigation, they sent representatives of neutral countries to meet leaders and regular people on both sides of the issue, including in Arab states around the British Mandate.

The UN proposed a partition plan. The plan would give the Jews, 1/3 of the population, around 55% of the area, though at least a third of that was just the Negev Desert, so empty and mostly barren. This was offset by land along the coast and some in the Galilee (north), which was fertile. The remaining 45% would go to the Palestinian Arabs. There would be an economic union, rules, and Jerusalem would be internationalized, so no one side would have control of it.

The Jewish leadership (minus a couple of fringe groups) accepted this proposal, but the Arab world lobbied heavily against it. When it passed the UN General Assembly, and amid celebrations by Jews, violence began to spiral out of control after an Arab militia attacked a bus with Jews (violence had already been simmering with tit-for-tat attacks for a long time prior). The spiral of violence turned into a civil war, with the Jews fighting the Arabs, and Arabs receiving support and money and volunteers from the surrounding Arab states. The Jewish forces had virtually won this civil war by May 15, 1948, the date the British had set for when their Mandate (and therefore responsibility for the area) would end, so the Jewish leadership decided to declare independence and fill the void. In response, the Arab states attacked. Israel won the war, and thus became a recognized country as the world grew accustomed to this setup.

During this period, Palestine had never been a state of its own. It did not become one, either. The Arab forces that invaded the area simply did not leave, because the Arab states wanted the land for themselves. Egypt ended up taking control of Gaza, while Jordan took control of the West Bank and what is now called "Eastern Jerusalem", which included the Old City and its religious sites. During this period, they chose not to give Palestinians a state of their own, and Palestinians continued to have the goal of destroying Israel and nothing less until 1988, when they began the process of renouncing this goal, formalized in the 1993 Oslo Accords.

Palestinians as a distinct group depends on what you mean. Were they a national group? Not fully. There were nationalist sentiments, but they were divided between pan-Arabism, pan-Islamism, and a variety of other identities. Palestinian nationalism began to develop but did not become extremely prominent until after the British gained control of the area.

Were they distinct administratively? No. What we today regard as Israel and Palestine, or what was once called the British Mandate for Palestine, was never its own province. It was multiple provinces, and the borders of those provinces shifted regularly.