(I used the names Tobi and Pain to avoid spoilers in the question, but herein I will use the character's actual identities, as it is necessary to avoid confusion. So, spoiler warning.)
Reviewing episodes 346-348 (World of Dreams, Creeping Shadow, The New Akatsuki), which tell the backstory of both Obito Uchiha and Nagato, I found no indication that Obito told Nagato about the plan to use Infinite Tsukuyomi. Obito, posing as Madara, speaks to Nagato several times and they argue; These episodes make their ideological differences clearer than anywhere else. Nagato, trusting in his friend Yahiko's ideals, says that the world can be saved by sharing pain. Obito argues that it's all futile, and that though they try to understand others pain, others do not try to understand theirs, and that peace is impossible in this world. Obito does allude to Nagato that with the Rinnegan, Nagato can actualize or create a world free of pain, but goes into no specifics. And when Nagato finally turns to his side, he seems to let Nagato form his own conclusions on how to do this, as Nagato basically forms his cycle of pain plan for world peace (at the end of The New Akatsuki).
It's difficult to say for certain Nagato didn't know the Tsuki no Me plan, and I can't go through all of the episodes to make sure, but I'm going to go off of the assumption he didn't, because that's what I'm seeing here. My conclusion here is that Obito saw no reason to get Nagato on board with Tsuki no Me, so long as he was able to align their interests in collecting the Tailed Beasts. It's likely he realized Nagato wouldn't be able to completely shake off the influence of Yahiko's idealism, and so merely sought to corrupt the whole idea of sharing pain. Considering awakening the Rinne Sharingan was necessary to achieve Infinite Tsukuyomi, and it seems unlikely Obito would have trusted Nagato to become the Ten-Tails Jinchuriki (he didn't trust Madara to do it, since he decided not to revive him), it is very likely he always planned on betraying Nagato. Thus, it makes sense to limit Nagato's knowledge of the plan. It's unclear how much Nagato discerned on his own and how much he trusted Obito, but it's likely he and Konan were suspicious (I think there are indications of this in the series, but I haven't had a chance to look back at them, but for instance, Konan was very much aware Obito would come after Nagato's Rinnegan after he died).
Another aspect to consider is that Obito's plan would have been very unappealing to the rest of the Akatsuki (referring to the rogue ninja Deidara, Hidan, etc., not Yahiko's original members, who Obito straight-up killed, but who probably also would have disliked the plan). I don't recall if the other Akatsuki even know what the true goal of the organization was, so I checked the wiki to jog my memory:
Although some members joined Akatsuki to further their own goals, all are united in Akatsuki's ultimate goal of gaining world domination. Their leader has mentioned a three-step process to work their way towards this goal. First, they plan to obtain a large amount of money to support their organization. Second, they plan to set up a mercenary group loyal only to Akatsuki. To build their reputation, they plan to accept any mission below the usual cost and operate off of the excess funds earned in the first part of their plan. Because there haven't been any wars in some time, the established shinobi villages couldn't afford to charge such low prices, leading the smaller countries to hire Akatsuki over the more expensive shinobi villages. By using the tailed beasts, Akatsuki would start wars and then immediately quell them, solidifying their reputation and convincing every country to depend on them, thus cornering the market in shinobi labor. Without any business, the other shinobi villages would collapse due to the lack of funds. Finally, after Akatsuki is the only major force of shinobi in existence, they can easily conquer the other countries. Members do not automatically learn of the plan upon joining; Hidan didn't find out about it until he had been a member for quite some time.
The plan told to the other members is far closer to Nagato's plan than Obito's. If they told the other members they were going to trap everyone in a genjutsu, including all of them, chances are good they wouldn't go along with it (people generally dislike being told they'll be trapped in a dream forever). But even Nagato's true plan might be appealing for many of the Akatsuki members, since, as the ones with the super weapon, it would put them at the top (they all already agreed to world domination). Thus, it makes a lot of sense that Nagato (as Pain) is the acting leader of the organization, while Obito (as Madara and Tobi) is the shadow leader, with his secret plans.
To review, the plans probably differ because Obito never told Nagato the actual plan. Obito likely did not trust Nagato with that information, and only sought to use him as a pawn to his ends. Thus, he allowed Nagato to come up with his own answer and never corrected him; so long as their interests were aligned, he didn't care.