What they have in common
Both are diamond, with essentially the same chemical composition, crystal structure and optical and physical properties. A lab-grown diamond is not an imitation stone.
Cut, colour, clarity and carat mean the same thing whichever you are looking at. How a particular stone behaves with light still depends on that stone — its cut above all, with its colour and clarity alongside — so two are not interchangeable merely because they share a category.
Where they differ
The difference is origin. One formed underground over a very long period; the other was grown in controlled conditions over weeks.
Four things follow from that, and they are worth keeping apart. Origin is where the stone came from. Identification is how a laboratory establishes which it is, and documentation is how that is recorded — a laboratory-grown stone is typically reported and disclosed differently from a natural one, and the report you are shown is not always the same document. Quality assessment is a third question, about that individual stone. Market pricing is a fourth, and it responds to supply, demand and how the two categories are traded, not to origin alone.
How the market treats them
Natural and lab-grown are priced and traded separately, and the two markets have behaved differently over time.
This is one area to be careful with: nobody can tell you what either category will be worth in the future. Buy the stone because you want to own and wear it, not on the basis of a prediction about later value.
Questions worth asking
Ask which category a stone belongs to and how that is documented. Ask what the accompanying paperwork covers and who issued it.
Ask how the seller handles each category in practice — documentation, warranty and any aftercare are set by the company you buy from, and should be confirmed with them directly before you commit.
Deciding
There is no universally correct answer. Some people care a great deal about natural origin; others would rather put the difference towards size, setting or simply keep it.
Both are legitimate choices. The one that matters is the one you will still be comfortable explaining in ten years.