The substances are not directly stated to be the same thing. This can be found on the notes section of the Black Liquid entry on Xenopedia:
In the first scene of Prometheus, an Engineer sacrifices himself by drinking a similar substance which causes his body to disintegrate and his DNA to spread to the seas of a primitive planet; implied to be Earth. It's not known whether the substance was the same as the Black Liquid.
Whilst they behave somewhat similarly the substance the engineer drinks is designed to break his body apart and split his gene code into a planet's ecosystem and therefore create the chain of reactions resulting in life. Whereas the goo discovered by the crew of the Prometheus when used on Planet 4 appears to cause a very different effect, it actually does cause some of the disintegrating engineers to develop chestburster's which can be seen in the flashback sequence. Furthermore David experiments with differing amounts of the liquid on several engineers to create different parasites, including the Neomorphs.
The black goo is also referred to as a bioweapon and a pathogen whereas the substance the Engineer drinks is displayed as neither. Whilst it does disintegrate and kill the Engineer that drinks the substance it does not begin creating parasitic organisms like that seen on Planet 4. When David unleashes the black goo we see the Engineers either disintegrated, infected with parasites or both. The engineers on LV-223 also seem to have had to deal with an outbreak of some pathogen which appears to be instigated by the black goo. That pathogen does not disintegrate the engineers. Whist it is entirely possible that the Engineers were killed by an organism like the Deacon (which itself is indirectly created by the black goo) if we examine the Engineer head taken from the head room.
It was sliced off by the door resulting in the Engineer's death and remained inside the head room with the numerous vials of black goo. We know that when the atmosphere of the room was disturbed the vials began to leak, given the room was unsealed it is possible that black goo residue was present in the room and equally possible the Engineer's severed head may have ended up lying in a pool of it with no specific effect occuring. When the crew of the Prometheus awaken the severed head it immediately begins to react in terror as its head bulges and eventually explodes which we see has been caused by the black goo. So we know the Engineer had been exposed to the liquid at some stage, seemingly before death but possibly not.
Instead of disintegrating, this Engineer instead reacted in a manner consistent with how the parasites created by the goo whether they be Neomorphs, Deacons or anything else tend to exit from their hosts body. By bursting out. There was no parasite within the Engineer head of course but when we examine the flashback in Covenant where the Engineer population on Planet 4 is subjected to the liquid we do see the parasites that proceed to immediately burst from some engineers are not specific creatures and it is actually the goo manifesting as Xenomorph like organisms bursting from the victims rather than actual parasitic organisms which develop later in other ways such as the spore sacks for the Neomorphs and the experiments of David.
Returning to the Engineer that seeds the planet implied to be Earth, there is no mention of the substance the Engineer drinks doing any more than splitting the Engineers genetic code which it reacts with and changes before entering the planets ecosystem. There is a strong likelihood that flora and fauna may already be present on the planet in some degree given the presence of water but the substance the Engineer drinks has no indication that it does anything to these organisms whereas the black goo we know for a fact had a direct impact on the native flora and fauna. David himself conducts various experiments on creatures native to Planet 4 whilst perfecting "his" perfect organism and the spore sacks that cause the Neomorphs seem to have appeared after the black goo mutated some of the native plant life, the parasite being a product of the goo whilst the plant being mutated into a sort of egg for the Neomorph spores.
Having said this the Xenopedia entries do tend to consider the goo and the substance to be similar for the most part:
The Black Liquid, as its name indicates, is a black amorphous compound of millions of microorganisms which behave as a single parasitic entity that can seemingly affect any faunal organism. The effects, however, are variable. Sometimes the Black Liquid merely kills victims, which it does by provoking nearly instantaneous deterioration of cells and tissues, resulting in a quick and gruesome death, as seen with the Planet 4 Engineers. On other occasions, however, the Black Liquid merges itself with its host and mutates it, altering both its appearance and its behaviour, the latter becoming far more hostile. Creatures affected by the Black Liquid may give rise to entirely new species, as demonstrated by the Hammerpedes and the spores that give rise to Neomorphs.
We know both substances cause the complete cellular disintegration of their victims however in the case of the sacrificial Engineer, the goo appears to go no further in effect than the sole Engineer whose body seeds the planet whereas the black goo we are more familiar with acts like a pathogen affecting all organic matter. We see it kill multiple beings via disintegration but it lingers and mutates other organic life forms unlike the substance drunk by the sacrificial Engineer.
We really won't have a more concrete answer until Ridley decides if he wants to tell us and it's really down to what you feel works more. The substance and the black goo do indeed cause very similar reactions in their victims however the black goo acts more like a pathogen whilst the substance acts more like a catalyst in a chemical reaction, it sparks a change on a cellular level and causes the affected body to break down and change. The black goo simply infects and kills its victim in a variety of gruesome ways.
One last thing to consider is also the manner in which these substances are exposed to their victims. The substance the Engineer takes is ingested in the same manner as a drink is taken into the body. The black goo is often seen coming into contact with the skin in the case of Fifield and the Planet 4 Engineers, presumably also the creation of Neomorph spore sacks is the liquid coming into contact with a plant. The Neomorph spores themselves embed themselves into their victims skin after entering an oriface such as the mouth or nose. The worms are also drenched in the liquid before mutating into the Hammerpedes. The only person who ingests the black goo in a manner similar to the Engineer is Charlie Holloway. Whilst the genetic difference between the Engineer and a human is likely to cause some variation in the effect such as the Engineer giving birth to a parasitic creature different to the Xenomorph commonly birthed from a human host the effect on Charlie is considerably different.
The Engineer that ingests the substance disintegrates at a cellular level. Charlie who inadvertently ingests the black goo begins to instead mutate into something else. His sperm mutates into a parasite that causes the trilobite to infect Elizabeth Shaw and moments prior to his death we see that his veins, skin and eyes are beginning to change. Earlier than that when he realises he's been infected Charlie notices a tiny worm like creature in his eye. Given both substances are taken in similar ways, through ingestion, which results in hugely different outcomes. The Engineer immediately breaks apart on a genetic level moments after ingesting his substance. Charlie doesn't break apart and parts of his cells mutate into entirely new species. This is more in line with the parasites than the Engineer's drink.
Given the similar DNA between Charlie and the Engineers, if the substances they ingested were the same thing then surely their reactions to the substance would be essentially the same. Even with Charlie ingesting a smaller amount of the liquid, the effect would basically be the same - disintegration on a cellular level. Instead Charlie begins mutating and would likely have ended up becoming something similar to Fifield though perhaps with a higher degree of intellect given Fifield sustained acidic burns to his face shortly before being infected by the goo, if the acid is anything like the Xenomorph or facehugger blood, there's a chance it may have severely damaged his brain before mutation, the goo itself even forces the acid on the glass into Fifield's screaming face before taking hold of him and the originally design gives Fifiled a more Xenomorphic appearance following his mutation. It is highly likely Charlie would have ended up the same as Fifield since they are both human and the goo does force itself into Fifield's mouth (amongst others) which is how Charlie ingests it.
Fifield also is affected by a large amount of the goo, which would probably equate to an amount greater than what the Engineer drinks and still Fifield does not disintegrate as the Engineer does. Therefore it seems quite plausible that while both substances are similar they behave in very different ways. After all it makes sense for the Engineers to not use a bioweapon as a means for seeding a planet and it is also later reasoned by Shaw and David that the Engineers were planning to bombard Earth with the vials of black goo in an effort to wipe out the population instead of create new life. Even with parasites emerging, the Engineers don't seem to regard these creatures as true lifeforms and would merely wait for them to die off after the hosts are expended on the planet. They would then be able to re-seed the planet after several millennia.