Both
The Spectre and
The Phoenix (aka The Phoenix Force) are nigh-omnipotent cosmic beings. Both can literally alter reality at will at any level.
There are not any differences in the cosmic entity called
The Phoenix. Marvel
nerfs the cosmic abilities of the entity whenever it inhabits a mortal being in varying degrees. The company likely does it to drive storylines for differing personalities.
That is why there appears to be different power levels and capabilities displayed by different characters whom the entity inhabits at any given moment. The entity itself always remains near all-powerful.
This is just my brainfart about the matchup. Both characters have been portrayed on paper differently but what each character is capable of doing are similar. That may not be apparent to some readers of
The Spectre because he is almost always shown interacting at the mortal level with other beings. In contrast,
The Phoenix has been shown throwing down repeatedly with other high level cosmic beings and intergalactic empires.
My first impulse is to easily go with
The Phoenix. What has stalled my thinking process is that usually (not always but usually) the cosmic entity co-inhabits a mortal being. On the other hand (and while trying not to break site rule #6

),
The Spectre is literally a divine being. The closest comparison I would make for him in the Marvel Universe is Eternity. He is not on the Marvel level of
The One Above All or
The One Below All.
So, my nutty mind is doing an inner death match between
The Spectre and
The Phoenix. I will cast my vote after either I see a TKO or a knockout.