If you eliminate would, you change the meaning.
The first means:
Not long after that, Perkin made the scientific breakthrough that (in the future) was going to bring him both fame and fortune.
The second means:
Not long after that, Perkin made the scientific breakthrough that (in the past) brought him both fame and fortune.
So, the two statements are looking at things from different points in time: The first is looking ahead to fame and fortune that (the writer knows) is still to come. The second is looking back on fame and fortune already achieved.