I realized 证 could be 証 and 證. Is there any difference between them?
In 繁體字 證 and 証 are two different characters. As stated in 教育部重編國語辭典修訂本 of Taiwan,
verb 1 告發。說文解字：「證，告也。」論語˙子路：「其父攘羊，而子證之。」 2 用憑據、事實來表明或斷定。如：「證明」、「證實」。莊子˙秋水：「證曏今故，故遙而不悶﹑掇而不跂，知時無止。」
noun 1 憑據。如：「人證」、「停車證」、「借書證」。 2 病症、症候。如：「三焦辨證」、「血分證」。列子˙周穆王：「遇老聃，因告其子之證。」
So if you are using 証 for its only rarely-used meaning of
admonish, remonstrate, then it cannot be replaced by 證.
In other cases 証 is used as an 異體字
(informal) variant of 證, acceptable but less preferable.
However, this post from Taiwan maintains that 証 is an erroneous form of 證 out of the reason of orthodoxy (despite the fact this variant is widely accepted).
Another post from HK gives more references. (It's written in Cantonese, though.)
In 簡體字 there is no such trouble since both of them are simplified to 证 (according to 新华字典).
From what I understand, 証 is a variant of 證 which means they essentially mean the same thing (see here or here). However, it seems like the usage of the two characters isn't identical. I can find no cases where 証 is accepted but 證 is not, but there are lots of cases where the opposite is true. All examples in the Ministry of Education dictionary are using 證, which wouldn't be the case if both versions were considered correct. Thus, I think it's safe to always use 證, even if both characters seem to be in use.
According to this site, it is acceptable to use 証 in most cases. A look at the etymology doesn't help much either. 証 seems to have had the meaning "admonish, criticise" originally, but I don't think this is relevant since 保証, 証物, 簽証 etc. all seem to be acceptable.
My conclusion: They are identical in modern usage. 証 is a variant of 證 and they can be used interchangeably.