Why unsafe state not always cause deadlock?
Deadlock means something specific: there are two (or more) processes that are currently blocked waiting for each other.

In an unsafe state you can also be in a situation where there might be a deadlock sometime in the future, but it hasn't happened yet because one or both of the processes haven't actually started waiting.

Consider the following example:

Process A                  Process B
lock X                     lock Y           # state is "unsafe"
                           unlock Y
lock Y                                      # state is back to "safe" (no deadlock this time.  We got lucky.)


There's a more interesting example in Section 7.5.1 of the link you gave:


  Consider a system with 12 tape drives with:


Process       Max Need       Current
P0:             10              5
P2:              9              3


This is an unsafe state.  But we're not in a deadlock. There's only 4 free drives, so, for example, if P0 does request an additional 5, and P2 does request an additional 1, we will deadlock, but it hasn't happened yet.  And P0 might not request any more drives, but might instead free up the drives it already has.  The Max need is over all possible executions of the program, and this might not be one of the executions where we need all 10 drives in P0.