Solution:
Both outcomes are equally likely! Notice that for every sequence with four heads (such as HTTHHTHTT) there is a corresponding sequence with exactly five heads formed by reversing all the coin flips (in this case $T H H T H T H H$). Thus the number of sequences of nine flips with exactly four heads is equal to the number of sequences with exactly five heads; since every sequence is equally likely this completes the proof.

In fact, both probabilities are equal to
$$\frac{\binom{9}{4}}{2^{9}} = \frac{\binom{9}{5}}{2^{9}} = \frac{126}{512}$$