I believe that some Age of Sail warships were in service for many years, so I suppose I mean a comparison between newly commissioned ships from the two periods.
From reading Brian Lavery's 2-volume work, Ship of the Line, over time there were some changes of note. In terms of sails, in 1715, there was a chance there being spirtsail topmast on the bowsprit (though even by that time they should have mostly stopped making those). By 1815, those were long gone. Over the course of the next century between those two dates, there would be plenty of work done with sails. Definitely more work with the sails that are on stays between the mast for fore-and-aft sails. The stay sails and jibs as so on (not to say those weren't used in 1715, but the ships of the line would know how to use them well by 1815 to say the least). The mizzen mast's lateen sail in 1715 would be gone and replaced by a gaff.
In crew size, the regulations as to how many men would be on each rating ship increased. The number of small arms and other crew combat weapons issued per the regulations doubled between the two dates (I remember that particular aspect from William Gilkerson's Boarders Away, another 2-volume publication). As for the guns, in 1715, brass (technically bronze) guns still had some significant presence in the Ships of the Line. By 1815, that wasn't the case. The carronade gun was introduced in the late 18th century, so by 1815 a Ship of the Line would have some of those guns too. For ignition system, in 1715, linstocks with burning slow matches lighting off powder at the touch hole on the breech of a gun lit off the great guns. By 1815, they actually had a flintlock mechanism that sat on top of the gun and could be fired via a lanyard that would set the mechanism off and create a spark to ignite the powder at the touch hole. The the guns themselves, the overall impression I got was that they figured out how to better balance calibers of guns to the decks they were placed. The earlier vessels tended to ride a bit top heavy (but it's not that clear if that was a vessel design issue or a gun weight issue), but that seemed to be a significantly downsized, if not resolved, issue by 1815.
For the lines of the ship itself - I don't remember terribly much difference, the big difference would be between a 1615 and a 1715 vessel, since from the sixteenth century to eighteenth century vessels stop building up their castles fore and aft and gradually bring them down and bring about a smoother run fore and aft. The grand stern carvings were already mostly gone by 1715, and a distant memory by 1815. The course of the eighteenth century is marked with making Navy/military decor simpler and plainer (unlike the seventeenth century).
Anything beyond that, I can't remember of the top of my head from reading that work I previously mentioned. There are many more intricate details that could be added, and maybe others here will.
The general external design would remain roughly similar throughout the period. There were some changes in hull shape, as it was worked out what was more efficient, but above the waterline, a ship from 1715 would resemble one from 1815. There were three major differences. Firstly, copper sheathing would cover the lower hull of a ship from 1815. This was first trialled in the 1750s, and introduced fleet-wide in the 1780s. This made ships faster by preventing the growth of weed on the hulls of ships. Secondly, a ship from 1815 would carry carronades on her upper deck. These were short-barrelled, cast iron cannon. They were lighter than an ordinary cannon firing the same weight of shot. As a result, carronades could be carried on the upper deck without a loss of stability. This greatly increased the short-range firepower of the ships. Finally, there was a change in the size of ships. The Royal Navy stopped producing the smaller 50-60 gun fourth-rates, and reduced production of third rates with less than 74 guns. The smaller sixth-rate frigates were also phased out, in favour of larger frigates. If we were to pick a random ship from 1715, it is likely that it would have been one of these smaller types, while the typical RN ship of 1815 would have been a large 30-36 gun frigate, or 74 gun third rate.
Hi there, I've answered some similar questions in the past. These links might be of interest, but please post follow-up questions if you have them.
http://www.reddit.com/r/AskHistorians/comments/2709jm/how_would_a_britishhms_frigate_built_in_1715/
http://www.reddit.com/r/AskHistorians/comments/20x9kz/classes_of_vessels_during_the_age_of_sail/