I know 党 on its own can be a suffix, but it can also be used on its own.
What is the difference between using 政党 and 党? Is one more formal than the other?
For example (using Goo examples): 党の方針 vs 政党の方針
党中に党を立てる vs 政党中に政党を立てる
I know 党 on its own can be a suffix, but it can also be used on its own.
What is the difference between using 政党 and 党? Is one more formal than the other?
For example (using Goo examples): 党の方針 vs 政党の方針
党中に党を立てる vs 政党中に政党を立てる
To some extent, yes, 政党 is more formal and 党 could be just a group. So 政党中に政党を立てる sounds less natural or just impossible while 党中に党を立てる could be used when there are sects in a party.
That said, 党 is mostly used when it refers to a specific party - in this sense, it more often traslates to the party. For example, 党の方針に従う is more natural than 政党の方針に従う because the party whose policy the subject follows must be clear from the context.
In contrast, 政党 refers to a generic party.
The former sounds inappropriate as a policy of the party (e.g., because it does not match what the party has advocated); The latter is more simply it is inappropriate as a policy of any party (e.g., because it is morally unacceptable).
This is similar for (one aspect of) the differences between 会社 and 社: 社の者 means a member of the company whereas 会社の者 is not equally idiomatic.
政党 is an unambiguous term that specifically means "political party", and 党 by itself is originally a word that just means "party (or faction, band, group, etc)". There are a number of words that contain 党 but have nothing to do with politics (eg, 私党, 徒党, 悪党, 残党).
The first sentence of a Wikipedia article on a political party almost always contains 政党 because this is the most unambiguous word. However, when the context is clearly about politics, saying 政 all the time can be somewhat redundant, so it's safe to just say 党 in the subsequent parts of the article. 党の方針 and 政党の方針 are interchangeable when you are clearly talking about a specific political party.
Although uncommonly, 党 can appear in the names of organizations unrelated to political parties. For example, Yellow Turban Army is often referred to as 黄巾党 in Japan. In fiction, 双蛇党 (Order of the Twin Adder) in Final Fantasy XIV is a military organization (or regime) rather than a political party, so it's a 党 that is not a 政党. Another example is 闇の一党 (Dark Brotherhood), an assassin syndicate in Skyrim.
(I'm not sure what you mean by (政)党中に(政)党を立てる... A faction in a political party is normally called 派閥.)