Rams were efficacious as a result of the type of construction used in the ancient world. A greek trireme was 50 tons of wood held together without a single nail or screw. The only metal in the entire thing was the ram. the ship was put together like this, held together with wooden dowels and rope and braced with some internal supports. This meant that (A) the side of the ship was relatively weak and (B) because all the structural strength was in the hull, if you broke the hull you had a pretty good chance of the whole enemy ship breaking up under its own weight.

Medieval (and possibly late roman) galleys were skeleton or frame built rather than hull built. That is, they were assembled around a internal wooden frame that carried the structural stresses while the hull just kept the water out. This meant that if you penetrated the hull, you could put a hole in the side of the ship, but you didn't destroy the structural integrity of the whole thing, so collapse was a lot less likely. This made rams a lot less deadly, so they stopped getting used. Above water "rams" were retained, but for boarding, not sinking enemy ships, and thus aren't properly called rams.

For more information on ancient greek ships, see Johnathan Hale's Lords of the Sea. For medieval and early modern galleys, see John Francis guilmartin Gunpowder & Galleys and Galleys and Galleons..

The other answers here have done a fantastic job of showing why, in general ramming as a naval tactic was not as popular in later periods. I'm going to point out that while it was not as popular, ramming ships were not completely phased out by medieval armies. In particular I am going to be using the navy of Alfred the Great as an example. In particular, it's important to know Alfred seems to have developed his navy in the context of a significant Danish/Viking threat. This is adapted from something I wrote earlier this year on Alfred's reforms, but hopefully suits your purposes.

The Anglo-Saxon Chronicle (ASC) reveals that Alfred commissioned the building of several langscipu, or long ships, to contest the Danes’ control of the sea.[1] These boats are attested to being twice as long as those of the Danes with as many as sixty or more oars each.[3] The same extract from the ASC denotes that these ships were a new model, being swifter, steadier and higher than other ships, and were not made after either the Frisian of Danish models.[4] It is particularly interesting that the ASC’s author mentions that Alfred’s vessels were not based off the Frisian or Danish ships, as these two peoples were the most adept seafarers of the North Sea in the ninth century.[5] This may be a nod to Alfred’s hiring of engineers from abroad to design and build these new ships, both Asser and Aethelweard noting particularly prominent Frisians, and potentially less prominent Danes, serving in the West Saxon navy.[6] While the chronicler does not explicitly say which type of ship Alfred’s vessels were superior to, their use of the word aescas to describe the Danes ships in the same passage indicates Alfred’s vessels were being compared to the Danish askr, a specific type of Danish craft largely made from ash-wood.[7] The askr was thus a flexible craft, and combined with its long, drawn out body, could make landfall both on coastal beaches and up river. A ship of this type is best preserved at the royal centre of Roskilde, wreck five being eighteen meters long, two and a half meters wide, one meter high and containing twenty four oars.[8] This style of ship was ideal for Danish purposes, be it ‘hit-and-run’ raids or for the swift transportation of merchandise to markets. As the askr was so well suited to Danish purposes, it may seem surprising that Alfred did not adopt askar of his own. Given Sidonius Apollinaris stated that the Saxons were once feared pirates themselves, it might seem odd that Alfred commissioned new ships at all.[9] Alfred may have actually used similar ships to the askar prior to 896, as the ASC records Alfred sending ships against a Danish force in 882.[10] Regardless, Alfred’s naval innovation appears to recognise two things; the first was that it was easier to stop raiders at sea before they made landfall, and the second was that the askr was primarily a transport vessel, not a war ship. While the askr’s length and short sides enabled swift transport, it also made it far easier to board during naval battles.[11] This is demonstrated well by Johnson’s reconstruction and subsequent sailing of the ships found in Skuldev, who concluded ships such as the askar were used primarily for their ability to travel long distances in a relatively short amount of time.[12] The West Saxon navy at this time was needed primarily for defence, not long distance journeys for trade or raiding, and Alfred’s new model of ships reflect this. While Alfred’s ships would have been harder to land on a beach due to their size, the increased amount of oars would have meant they could out speed an askar. This would have been particularly useful for ramming the raiders boats, a technique that flexible ships akin to the askar seem particularly vulnerable to.[13] Swanton has gone so far as to argue that as the Old English word keel was often equated with the Latin rostrum, or “beak”, it is likely ramming was a common naval tactic of the Anglo-Saxons.[14] This technique’s effectiveness was seen in Gifford’s re-enactments of naval ramming using reconstructions of both ship types.[15] Thus it seems that ramming was still an effective naval tactic in the medieval period, at least in the context of Alfred's Wessex.

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Ramming ceased to be the main method of attack, but was still used from time to time when the situation called for it.

Famously, at the Battle of Lissa (1866) the Austrian navy relied heavily on ramming to best the Italians. This action lead to a increase in interest in ramming, and the development of a small number of ships designed largley around the ram - especially the idea of coupling a ram with the slow/inaccurate torpedoes that had recently been developed (see, for example, the torpedo ram HMS Polyphemus or the harbour defence ram USS Katahdin).

While the idea never took off, that dead end of ship design is forever remembered in one the most famous fictional warship of all time - the torpedo ram HMS Thunder Child from H.G. Well's classic The War of the Worlds.