第一个是对抗，第二个是抵消。比如第二个的语境可以是爸爸给我的支持offset了一部分妈妈给我的压力

I would never put thee words side by side. You need to put each in context. Counter is usually used to mean 'in direct opposition'; offset to mean 'having a balancing effect' ; however it is much better to see sentences where these appear, and let me assure you they are not synonyms.

Counter is the complete reverse of something while offset is only a partial reversal or sometimes just a sideways movement depending on context