两者均可表示“一天天(地)”，但含义和用法均有区别：
day by day 通常用作状语，且含有逐渐(gradually)转变的意思。
day after day 可用作状语，有时也可用作主语或宾语，它通常暗示时间之长久(for many many days)。
比较：
Things are getting worse day by day. 情况一天天变糟。
He is getting better day by day. 他一天天地好起来。
Day after day we waited for news from Sally. 我们一天又 一天地等着莎莉的消息。
Day after day went by, and still no message arrived. 一天天过去了，还是一直没有消息。
