I'm less sure about Qatar, but Bahrain was always a bit problematic within the original nine-emirate concept. It was larger, population-wise, than most, if not all the other emirates, and it was not particularly wealthy. So some of the other emirates were concerned that Bahrain would be a net drain on their resources.

There were reservations and rivalries between the nine emirates that made the nine-state union unlikely. I recently read an British archive document regarding a Bahraini delegation visit to the UK. The British resident in Bahrain advised ministers at home on the following: Bahrain's rulers felt that they had been bad-talked behind their back by the other Emirati states and would be trying to set the record straight; and they were advised to avoid answers questions regarding previous Emirati opinions voiced to them about Bahrain (as they were negative). Qatar and Bahrain had a long-standing rivalry, and territorial claims over the Hawar islands were only settled by the International Court of Justice in 2001.

Iran's claim on Bahrain also made the emirates with strong relations with Iran uncomfortable. In 1968, Iran warned that if the UAE declared independence and included Bahrain, it would oppose them, and that if Bahrain became an independent member of the UN, it would leave it. A large part of Bahrain (and the UAE's) independence depended on negotiating an end to the Iranian claim on the islands.

This finally happened in 1970 following a UN-mandated vote (where Bahrainis overwhelmingly voted for independence). Iran formally dropped its claim. (Interestingly, the Shah had been looking for a way to drop the claims without losing face at home, as the public was very much for pressing the claim). By the summer of 1971, the relations between the two states had considerably warmed, with diplomatic trips promoting peace between them. In August 1971, Bahrain declared independence and soon joined the UN (Iran congratulated them and didn't leave the body).

I am however not sure what led to the final break between Bahrain and the Emirates. It occurred in early 1971 (Saudi Arabia was against Bahrain breaking away from the emirates, and it took them some time before the Bahraini leadership felt prepared to "brave" the Saudi king's displeasure; at the time their economy was heavily reliant on Saudi Arabia). Britain played a leading role in preparing Bahrain's economy, courts, security apparatus and state structure for independence. This was by no means perfectly done, but helped give Bahrain's rulers the confidence to go at it alone. My impression is that Britain felt from the beginning that Bahrain as part of the UAE would be impractical.

Indeed Bahrain's history and state structure was different from the rest of the emirates. It was geographically the most distant (relative to the other emirates, which border each other) and it was the seat of the British colonial residency. Its politics were also much more complicated, with very active and occasionally violent leftist movements (particularly the Popular Front for the Liberation of the Occupied Arabian Gulf) at the time. The other emirates may have understandably seen it as politically unreliable.

UK Public Record Office, FO 1016/898, 1971

PRO, FCO 8/1643, 1971

Roham Alvandi (2010) Muhammad Reza Pahlavi and the Bahrain Question, 1968–1970, British Journal of Middle Eastern Studies, 37:2, 159-177

Sorry for the vagueness in my answer; unfortunately don't have access to some of my books and archive material today to expand on this any further today.

In addition to /u/i_like_jam's fantastic answer, I would point out that there was a third split of the Trucial States when Ras Al Khaima (RAK) briefly held out from joining with the UAE as well.

Like Bahrain, the key issue was Iranian claims to some of Ras Al Khaima's Persian Gulf Islands. For context, remember that the Trucial States were not just an imposition by Britain on the emirates, it also formed a set of treaty obligations on Britain to protect the emirates interests and sovereignty. So when Britain decided in 1968 that it was pulling back from "east of Suez" as the famous phrase goes, the prospect of independence without British military protection was not necessarily a happy one for the tiny gulf emirates.

As /u/i_like_jam mentions it was not until the the disputes with Iran were resolved that Bahrain felt it could safely declare independence. The same was true of RAK which held out from joining the UAE, thereby forcing Britain to theoretically uphold its treaty obligations pending independence. Unlike Bahrain, the solution for RAK, negotiated by the British, was to make direct concessions to Iran in the form of, in theory, half of the Tunbs Islands. This was similar to a slightly early agreement made with the emirate of Sharjah over the island of Abu Musa. The agreements were signed, RAK joined the UAE, but ultimately the Iranians seized the entirety of both islands.

Source wise, he only covers it in two pages, but as always Eugene Rogan's The Arabs: A History is a great.