How come the stock market crash of 1987 didn't seem to affect daily life like the crash that led to the Great Depression did ?
A stock market crash does not always lead to recession. The problem is that you assume that the GD was caused by the crash, while in reality, the real Great Depression only started after the stock market crash.
It is widely accepted in the academic community that the monetary system was the problem. In 1987 the central bank did the 'right' thing and nothing happened beyond that.
I will not go much further in this answer, If you have more detailed question please ask specifically.
Sources:
What Ended the Great Depression? by Christina Romer
A Monetary History of the United States, 1867-1960 - Friedman and Schwartz