It depends entirely on what you mean by 'the nature' or 'quality of of the carbonation'.
If we're talking carbonation and only carbonation (literally the level of dissolved carbon dioxide in the finished product, usually expressed as volumes of CO2/volume of liquid, or as parts per million) there's no difference between reaching that by bottle-conditioning or force-carbonating.
However the procedural differences between forcing and conditioning can affect the quality of the finished product in a number of perceptible ways.
When force-carbonating, CO2 of the highest purity must be used. Extraneous gasses (most onerously oxygen) or flavor/aroma compounds might otherwise be introduced and subsequently affect finished beer quality (though nitrogen is sometimes added as part of the profile of the finished product). In bottle-conditioning, CO2 evolved through yeast fermentation is chemically pure, and in fact the yeast growth in the early phase of re-fermentation scavenges oxygen from the beer and tends to improve stability. So you end up with a different profile of dissolved gasses in the end product.
If the beer is force-carb'd by injecting CO2 at high pressure into the liquid and periodically bleeding it off, flavor and aroma compounds (especially from hops) can be carried out of solution with the gas. You often hear of people bottle-conditioning dry-hopped beers to maintain the highest level of hop aroma. Conversely this technique could be a boon, for example, for a beer with a low hop aroma profile that ends up with too much sulfur dioxide, which can be mitigated by stripping with inert gas (CO2, N2).
It's also worth noting that this technique and other force-carbonating methods that involve lots of foaming (cranking up the CO2 pressure and shaking a corny, for example) can negatively affect the head retention qualities of a beer. Foam-positive agents that form as foam during processing (before it's in your glass) can either stick to the side of the vessel or form on the surface and return to the beer as 'flakes'. Any foam-positives lost during processing no longer contribute to foam potential in the glass.
-Binding of CO2:
According to Wolfgang Kunze in Technology Brewing and Malting, 'the binding of the CO2 (solution) is at first only loose. Permanent binding to the colloids in the beer takes some time'. Vague, yes. But it suggests that a very recently force-carbonated beer would show different characteristics in the way it foamed/lost carbonation.
-Presence and quality of yeast/other stuff:
There are many ways to bottle-condition that will give you a different outcomes: a fully fermented and conditioned beer can be filtered, then yeast and fermentable sugar added right before packaging (meaning flavor/aroma/mouthfeel/foam could be affected by what's filtered out, and the flavor/aroma contribution of the re-fermentation is more predictable because quantities of yeast and sugar are known); unfiltered, fully-attenuated beer can be dosed with fermentable sugar (meaning a possibly unknown quantity of yeast in the final product, plus less predictable flavor/aroma contributions by the re-fermentation due to the unknown physiological condition of the yeast); much less commonly, still-fermenting beer can be packaged (or at least held under pressure) when it is a known degree above final gravity, trapping the still-evolvoing CO2 in solution (meaning flavor/aroma compounds such as hop oils and sulfur compounds that would otherwise be carried out of the beer by escaping fermentation gases will now end up in the finished product, changing its quality).
Not to mention any conditioned beer has yeast in the final product which, depending on the amount and quality, can lead to quality issues due to autolysis.
"In the above passage, all the qualities attributed to natural conditioning could also be the result of slightly higher level of carbonation compared with the force carbonated beer."
Mostly, but probably not 'the beer feels smoother', and 'fewer bubbles sticking to the glass' would not indicate higher carbonation (though it sounds like that glass was probably objectively cleaner than the other).
To sum it up, while precisely the same levels of carbonation can be achieved either way, so many small things can be affected by how you carbonate that you'll never end up with exactly the same finished product.