What's the difference between 証 and 證?
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,


  證
  
  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。」


and


  証
  
  verb
  諫正。如：「証諫」。戰國策˙齊策一：「士尉以証靖郭君，靖郭君不聽。」
  
  =>「證」的異體字（07937)


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).