Back when I was a kid, car batteries used to be huge heavy lumps of plastic filled with lead and acid. They used to weigh almost as much as a mobile phone (slight exaggeration there, sorry).
45 years later, car batteries still look the same and weigh the same.
So, in this modern age and emphasis on fuel economy, why do batteries still weigh 40 lb? Why have advances in technology not been able to make them lighter and more efficient?
So, now after the answer to your literal questionto your real question, that you sadly didn't ask
The NiCd batteries are simply worse in every aspect but energy density than lead acid. NiMH is better, but much more expensive, and still has a higher rate of discharge, typically (unless you make them even more expensive). And still pretty hard to properly dispose of.
Lithium batteries aren't that easy handle. You need to protect them against all sorts of failures, and some of them are pretty fatal: don't overheat your lithium Battery. It will explode. And heat is a serious problem inside a motor compartment (in fairness, a battery doesn't have to be in there, but it's pretty handy).
The main reason really is cost. The battery in my last car, a 1999 Fiat Punto, supplied max 100 A (when I tried to estimate the actual short circuit current, around 43 A, but still a lot. Let's say P=U·I=12V·40A=480W) current, and had a nominal capacity of around 30 Ah (that's an energy of 12V·30Ah = 360Wh). It cost me 25€. So, rough guess, it's cheaper than 10€ to produce.
So, let's take a lithium battery type that is mass-produced and hence cheap. The commonly found round cells that make up many laptop battery packs are around 3€ each (let's say 1€ in production) for around 3Ah (11.1Wh), supplying up to 5A (tops, don't do that for long) at some 3.7 V. That says a single cell of these can supply 18.5W. So to reach the estimated 480W of my cheapo car battery, you'd need 26 of them. They'd cost 26€ in production, not counting the Euros you spend on control, charge and protection circuitry, on encasing them in something rigid and safe, and the fact that the minerals needed to produce some of the rare-metal components in Lithium batteries aren't currently getting cheaper, and equipping cars all over the world with those will definitely speed up that market mechanism.
Let's assume cost scales with capacity. My 26-cell lithium battery has 26·11.1Wh=288.6Wh energy. So we need to scale that by 1.25 to achieve the same 360Wh as the lead-acid battery.
Such a cell weighs around 90g. So the weight of the cells is 26·90 g = 2.34 kg. Ok, I don't have the exact weight of my cheap car battery in my head, but let's say it was 15 kg. So we saved weight by a factor of about 6.3, if our casing, and electronics are lightweight (they're not – as far as I can tell, you'll need a hefty switch mode power supply to be able to efficiently charge these using your car's generator, and those mainly consist of a pretty bulky coil of copper, and maybe some ferrite core that isn't exactly lightweight, either).
That leads to a cost factor of about 3.5 between component A and component alternative B, with handling disadvantages, lesser reliability and supply chain changes. No wonder the car industry isn't pushing in that direction. (And, by the way, they have excellent lobbying.)