Assuming that the connections are the same (which your post suggests is true), the only difference between the two adapters appears to be their current rating.
The PA-9 is rated as supplying 20V at 4.5A
The PA-6 is rated as supplying 20V at 3.5A.
The 'current rating' of an adapter shows the maximum current that it can supply without overheating. It can happily supply a lower current without any problems.
So, if a laptop was designed for use with the PA-9 adapter (because it 'draws' 4.5A), using a PA-6 would not be appropriate because the adapter would be required to provide a current above its rating (which could lead to overheating and potential 'burn out' of the adapter)
However, if a laptop is designed for use with the PA-6 adapter (because it 'draws' only 3.5A), a PA-9 adapter could be used without any problems (because the current 'drawn' by the laptop is lower than the maximum available from the adapter).
If I've read your question the right way round, you're seeking to replace a PA-6 with a PA-9. As long as the connections are genuinely the same (and not, for example, with the positive and negative connections reversed, which would seem to be extremely unlikely), there should be no problems.
If your laptop is failing to accept power from either adapter, it suggests a problem with the laptop's power circuit, rather than with the adapters.
Chris