This is quite a difficult question to answer as in the early 18th century the individual design of each frigate would have been down the the master ship builder at each individual dockyard. The idea of building to class doesn't come in until the 1740's, when ships began to be built under contract.

This means a great deal of variation between ships of the same rate. One of the biggest changes one would notice is that a lot of larger earlier frigates (5th rate) had their guns split across two levels. This meant their firepower was massively limited in rough seas, as the lower gun ports would be to avoid flooding. Later frigates put all the guns on the same level to avoid this problem.

The biggest leaps in frigate development for the British came after the capture of the French frigate "Hébé" in 1782. They initially copied the design, creating the Leda class of frigates. Which were faster and more manoeuvrable than the previous classes of 5th rate British frigates. The problem with them was that they had limited storage and were not very good in bad weather. The British took this design and developed the Lively class of frigates, taking all the pros from the French design and making the ships more suitable for long voyages and bad weather.

There is obviously a lot more to cover on this topic, but I have one Robert Gardiner book to hand the rest are back in Canada.

Reading:

R. Gardiner "The First Frigates" 1992

R. Gardiner "Frigates of the Napoleonic Wars" 2000

R. Winfield "British Warships in the Age of Sail" 2008

R. Winfield & D. Lyon "The Sail and Steam Navy List, 1815-1889" 2004"

/u/on1879 offered a good overview, but I'd like to add some more information that might contribute to a more complete understanding of the question. (Great question, by the way!)

It's important to note that the term "frigate" had different meanings at different times in the British navy. During the civil wars of the 1640s, the parliamentary forces fought mostly with armored merchantmen and did not make much use of the "great ships" that Charles I had built. After the third civil war started in 1649, English shipping was attacked by fast cruisers from Dunkirk and other Channel ports. The Dunkirker ships were called "frigates," but were very different than what we think of as classic frigates of the Age of Sail; they were very small, lightly built, with very minimal armament and large crews. They were designed essentially for commerce raiding and high-speed operations against weak targets.

The word, though, became applied to bigger ships that the English navy would build, and also became a verb (in French, "to frigate" meant to build long, low ships) and an adjective, so it loses a bit of descriptive meaning. In any case, in 1650 the English navy launched the Speaker, a two-gundeck ship originally built to carry 44 long guns and carrying 60 by 1655. The Speaker is arguably the ancestor of the two-decked ship through the 1750s.

The reason why I'm bringing this all up is that the shift from the shorter, handier ships of the Elizabethan era to these longer, lower ships with broadside armament meant a change in tactics, where ships would no longer approach and fire off all their guns before withdrawing, but would rather lie in line of battle and batter each other.

The drawback of building ships to this model is that the second (and in some cases, third) gundeck led the English to over-arm the ships, which made them top-heavy. This led to problems of the lower gundecks being submerged in any kind of weather, and also the ship heeling and not being stiff enough to get the full driving force out of its sails. New ships were sometimes "girdled" with extra planking at the waterline in an effort to increase buoyancy; this did not improve sailing qualities. These defects were serious in the North Sea and worse in the Atlantic, and were compound when Parliament became involved in ship design and voted in the 1690s money to build ships to an (unrealistic) specified tonnage and gun rating (80 guns at only 1,100 tons), which proved to be inadequate in practice.

To get to your question more specifically, it's a bit of a false premise. Parliament did not vote any new funds for shipbuilding between 1696 and 1745, so any "new" construction during that time would have been done under an administrative dodge of a "great rebuild," in which ships were more or less broken up and some of their timbers used in new construction. So a "frigate" built or "rebuilt" in 1715 would be of that older, two-decker type and thought of as a line-of-battle ship, so the ships "built" in 1715 were really of the older design dating back to the 1650s and later.

Now, technical improvements: /u/unclebourbon mentioned coppering. Copper sheathing replaced earlier experiments with lead sheathing that had been conducted starting in the 1670s. These failed not because of anything wrong with the premise, but because of electrolytic corrosion, which was not underwood at the time (the lead and iron parts of the ship combined with seawater to form a slight current which ate away at the iron).

A more obvious technical development of the 1690s was the invention of the steering wheel, which replaced the whipstaff attached to the ship's tiller. It doubled the potential range of movement of the rudder and allowed for many more men to combine their efforts to steer a ship in heavy weather.

Now, to get to the real meat of your question, which is frigate design between 1715 and 1780: it's complicated. I'm going to have to gently disagree with /u/on1879 on one point: the British did not copy the Hebe (or other ships); even if they had wanted to build exact copies of ships or even if shipwrights told their superiors they had built exact copies, British ships were always more heavily timbered than French ships and fastened with treenails, not iron nails. There were also substantial internal differences due to the British preference for increased stowage, and the decks/hatchways/magazines/etc. would have been changed to meet those needs. It's much more correct to say that British "true frigate" design was influenced by the capture of the French Medee in 1744, but that British fridges were built with British needs in mind. (The Medee was lightly built, like many French ships, and was not bought into service.) The shipwrights of the 1740s were comparing French designs with older British designs, such as the yacht Royal Caroline, and were building cruisers with 11-12 gun ports per side. These were initially built with a 22-24 gun battery of 9-pounders, and became larger over time, with 28-gun 9-pounder frigates being replaced by 32-gun 12-pounder frigates and the very large 36-38 gun 18-pounder frigates by the 1770s, in time for the first American wars.

To put those changes in perspective, it's useful to examine the change in armament of the ships. A 22-gun 9-pounder frigate would throw a broadside weighing 99 lbs (11 guns/side * 9 pound balls). The 32-gun 12-pounders would throw almost twice as much iron per broadside (192 lbs), while the 38-gun 18-pounder would throw 342 lb at once. All these ships were built with the idea in mind that they would be fast, weatherly and carry enough stores for a long cruise.

edit for sources: N.A.M. Rodger, Command of the Ocean; Ian W. Toll, Six Frigates; Robert Gardiner, First Frigates.

One change that would have been implemented in this time period that came to mind was the implementation of copper bottoming of warships.

Experiments on this began in 1761, and was eventually installed on all warships as a result of wars with France in the 1780's. The process involved nailing sheets of copper onto the hull of warships under the waterline using iron nails. This prevented the corrosion and biofouling of the wooden hull.

The benefits of this was that British warships were able to be kept at sea for longer periods of time before requiring maintenance in dry dock. I've also heard that it may have also given a small increase in ship speed as a result of a more streamlined hull but I can't find anything to back that up.

As for paint schemes, Nelsons Chequer scheme became quite common for British warships after his Victory at Trafalgar.