I'll answer this from regard to the British viewpoint, because I am by no means an expert on Napoleon.

Something to keep in mind at the start, though, is that Nelson's and Napoleon's careers are often set up as paralleling one another, when in fact Nelson was killed in 1805, during the War of the Third Coalition. At the time, Napoleon had only been Emperor of the French for about a year, but his ideas and personality were well known to the British.

Nelson's rise to prominence -- at least in how I would describe it -- was as a national hero as the victor of the Nile in 1798, which destroyed the French fleet in Egypt but did not capture or defeat Bonaparte there; later, Nelson remained a national hero especially after the battle of Trafalgar, but he was a different kind of hero/was differently prominent to the British. (Basically, I find 1:1 comparisons of the two men difficult.)

To answer your question more directly, Nelson was captain of the Agamemmnon at the siege of Toulon, while Napoleon was a young artillery officer.

The quotation you're referring to first appeared in Robert Southey's Life of Nelson, published in 1898.

So there are two ways to view this: First, to say the least, Southey's biography is fully a product of the Victorian great-man school of history, as you can see by glancing at this excerpt from the introduction (sorry for the format, Google Books won't allow copying and pasting). It would be well within the model of the day to ascribe prescient views to Nelson where possibly none existed.

Second: The context for Nelson's letter to his superior that carried that quotation was the activity on the Italian riviera that started in 1795 and continued on into 1796, when Bonaparte first pushed actions from a relatively junior position in Paris, then took personal command of a French army fighting both Austria and Sardinia-Piedmont, eventually capturing Milan. Nelson was in charge of a squadron meant to give relief and supplies to the allied armies, but which was hamstrung with a lack of resources, the ongoing neutrality of Genoa, and the loss of Austrian ports during the campaign in 1795. The event that sparked the letter to Jervis was the capture of a brig laden with materials for the siege train at Mantua, including letters and military texts meant for Bonaparte. So it's entirely possible that Nelson's reaction was simply grumbling about the French commander, who was in any case making a name for himself in this campaign.

For a more nuanced view of Nelson's view of the French in general, we can turn to a more recent biography of Nelson, John Sugden's Nelson: A Dream of Glory. (This title is the first of two by Sudgen that covers Nelson's career, but speaks to the time period you're asking about.)

I'll quote at a bit of length:

Nelson had always disliked the French, but it was the revolutionary wars that turned his prejudice into a corrosive hatred. He began them by doubting the wisdom of intervention, though from a personal point of view he was grateful for the employment. Convinced that the revolution would destroy itself, he actually believed his country’s ‘wisest’ course was to stand back and watch it happen. But the appalling atrocities “the republicans visited upon their own dissidents – anti-centralists, anti-Jacobins and monarchists – sharpened Nelson’s hostility. He witnessed the fates of the defeated Royalists of Toulon in 1793, and in 1795 and 1796 saw the devastation Bonaparte’s pillaging hordes created in Italy. French imperialism was not invariably destructive. It introduced some benign reforms and adopted anti-denominational rather than anti-religious policies, but nowhere was its capacity for plunder more obvious than in Italy, where French armies lived off the land and a sum of fifty-three million livres was levied on the occupied territories before the end of July 1796. Nelson’s determination to spare Britain similar ruin increased his ferocity. ‘To serve my king, and to destroy the French I consider as the great order[s] of all, from which little ones spring,’ he wrote in 1799. ‘And if one of these little ones militate against it . . . I go back to obey the great order and object, to down, down with the damned French villains. Excuse my warmth, but my blood boils at the name of a Frenchman. I hate them all – Royalists and Republicans.'

Nelson’s extremism was stimulated by fears, not so much from any regard for French arms, for which he cared little, but from the more invidious infiltration of their political principles. For the wars that broke out were not in the end just dynastic and national disagreements, territorial squabbles and commercial and imperial competitions, although they were in part all five. They were also ideological conflicts fought for minds. Monarchism, Christianity and self-determination were, in Nelson’s view, pitted against republicanism, atheism and imperialism. ‘It is their infernal principles I dread, not their prowess,’ he remarked. He knew that those ideas, cloaked in the language of liberty, popular sovereignty and natural rights appealed to every suppressed group. They were capable of slicing through ancient orthodoxies as a hot knife through butter, and of infecting and destabilising England without a single French soldier landing on her soil.

Nelson and Bonaparte are often compared as commanders, and the two men had deeply different personal styles. I should say at the outset that any comparison is likely to be flawed, because war at sea and war on land are different (the old problem of the elephant and the whale), and many people have forced them into contrast with one another. Painting broadly, Napoleon's attitude towards his subordinates tended to be distrustful, and he tended to exercise centralized control over battlefield decisions and to leave his subordinates clueless as to his intentions. (Historians still argue over the degree to which this affected his performance at Waterloo.) Nelson, on the other hand, grasped that centralized control of a sea battle, which had to be carried out by flag, lantern and/or gun signals, was a tricky affair at best, and so he did his best to communicate intentions to his captains by meeting with them often and informally in ways that valued them in the discussion but made his overall plans clear. To quote from Sudgen again:

From the beginning he involved his captains in the command process through a series of informal meetings in which strategy and tactics and ways and means were thoroughly aired. As later described by Berry, during these assemblies the admiral’s practice was to ‘fully develop . . . his own ideas of the different and best modes of attack, and such plans as he proposed to execute upon falling in with the enemy, whatever their position or situation might be, by day or night’. In other words he discussed what might be done in every eventuality. ‘There was no possible position in which they [the enemy] could be found that he did not take into his calculation,’ and for which he did not suggest ‘the most advantageous attack’. Because the captains had been comprehensively briefed, understood Nelson’s intentions, and had examined the different ways of achieving them, ‘signals became almost unnecessary’ and ‘much time was saved’. Each captain was theoretically capable of using his own initiative to achieve the corporate end.