For making sense of the Shia claim in reference to that verse, you should take into account two important factors:
- The hints that exist in the actual Arabic version of the verse that help us identify its different implications.
- The historical context for the verse.
The Arabic version of the verse is composed of three statements:
- وَقَرْنَ فِي بُيُوتِكُنَّ وَلَا تَبَرَّجْنَ تَبَرُّجَ الْجَاهِلِيَّةِ الْأُولَىٰ ۖ
Stay at home, and do not deck yourselves with ostentation as in the days of paganism;
- وَأَقِمْنَ الصَّلَاةَ وَآتِينَ الزَّكَاةَ وَأَطِعْنَ اللَّـهَ وَرَسُولَهُ ۚ
fulfil your devotional obligations, pay the zakat, and obey God and His Apostle.
- إِنَّمَا يُرِيدُ اللَّـهُ لِيُذْهِبَ عَنكُمُ الرِّجْسَ أَهْلَ الْبَيْتِ وَيُطَهِّرَكُمْ تَطْهِيرًا
Indeed God desires to remove impurities from you, O inmates of this house, and to purify you a thorough purifying. (Quran, 33:33)
If you know Arabic, you will notice that the pronouns used in the first and the second parts of the verse are female pronouns, whereas the last part --i.e. the part used in support of Shia belief-- uses male pronouns. Shias highlight this alternation in Grammatical form as an indication that this verse is talking about two different groups within the family of the Holy Prophet, i.e. 1) Prophet's wives as in the first two parts and, 2) a specific group within Holy Prophet's family that have been subject to the mentioned Divine act of purification that comes in the last part.
This is the argument based on the semantic analysis of the verse alone. But Shias also put forward historical narrations to highlight what they see as the actual implication of the verse. The narration is called the Hadith of the Cloak (or Hadith al-Kisa or Ahl al-Kisa) and it shows up in both Sunni and Shia sources. Here's an example from Sahih Muslim:
'A'isha reported that Allah's Apostle (may peace be upon him) went out one norning wearing a striped cloak of the black camel's hair that there came Hasan b. 'Ali. He wrapped hitn under it, then came Husain and he wrapped him under it along with the other one (Hasan). Then came Fatima and he took her under it, then came 'Ali and he also took him under it and then said: Allah only desires to take away any uncleanliness from you, O people of the household, and purify you (thorough purifying) (Sahih Muslim, Book 031, Hadith 5955)
As you can see from this hadith, the context for the last part of the verse has been an incident when Prophet had gathered only a specific members of his family under his cloak and then declared them to be the ones purified. This specific use of purification is also interpreted in moral and spiritual terms for the clearly emphatic rhetoric of the last part of the verse, through use of the adverbs "indeed" (أنّما) in the beginning and "تطهیرا" -- which can be translated to "thorough or full purifying" -- in the end.
The argument for infallibility of these specific members of the Holy Prophet's household is further consolidated by evidences from hadiths. For a deeper study you may want to look at Wikipedia entry on Ahl al-Kisa; Ismah (infallibility) and this article from al-Islam.org titled Ahlul Bayt (`a): Its Meaning and Origin.
As for the mention of "purification" in other verse you quoted, that verse is talking about a different notion of purification and in a specific and limited context. It is in reference to the downfall of rain on a number of Sahabas who had a wet sleep the night before, and were thus purified from their bodily impurity. Plus, it makes no implication of "thorough purification" which can be interpreted as suggesting infallibility.