The latest batteries are much lighter and cost less over a vehicle lifetime than ones of yore. But they do not use LA (lead acid) chemistry.
A LiFePO4 (Lithium Ferro Phosphate) battery will do what is required at acceptable whole of life cost BUT at higher initial capital cost - which makes it unattractive to car manufacturers.
Low initial capital cost seems to be the main reason to prefer lead-acid to LiFeO4 and it's not obvious that there are any other really good reasons.
Cycle life is very much greater than that of Lead Acid, which allows whole of life cost to be lower than lead acid.
Unlike LiIon (Lithium Ion) a "spike through the heart" will not cause the issues a LiIon has.
Charging control is "easy enough".
Compared to lead-acid:
Allowed depth of discharge, & max acceptable charge rates are higher,
Temperature range is better
Recharge efficiency is better.
Self discharge performance is better.
Lithium Ion / LiIon:
It's worth commenting on LiIon batteries as they often get "bad press" with respect to safety.
Compared to lead-acid, LiIon chemistry offer substantially better mass and energy densities (lighter & smaller), somewhat longer cycle life, higher capital cost and probably somewhat superior whole of life cost. Properly managed, charging control is easier. Temperature ranges are better, charge/discharge efficiency is somewhat superior. Disdavantages relating to safety are largely not an issue - see below.
In many applications LiIon batteries are the battery of choice - from Dreamliners to Samsung phones to "Hoverboards", Mars Rovers to laptops and smartphones to MP3 players and more. The first three applications above were selected for their known spectacular failures. But anything used in a Mars Rover is chosen for its suitability in a long life, hostile environment, must not fail task. And there are hundreds of millions of LiIon batteries in everyday use in people's pockets and homes and cars and more.
Given the ways in which LiIon batteries CAN fail, the numbers that DO fail in a spectacular manner are very rare. Failures that are widely reported are quite often due to some systemic failure that affects a batch or model of battery that has been produced and distributed in vast quantities OR lower volume bu high profile applications. In such cases a design or manufacturing fault or shortcoming causes or allows failures whose consequences are exacerbated by the LiIon chemistry's unforgiving behaviors.
Examples are well publicised "vent with flame" events in some past Apple laptops, Samsung phones, self-balancing "hoverboards" and similar. In the 1st two examples usually competent manufacturers allowed a design fault to exist uncorrected and/or unnoticed or cut corners in manufacturing to the extent that safety margins caught up with them. In the case of the "hoverboards" the cause is unknown to me but is as liable to be low quality low cost manufacture and poor charge control as anything else. In consumer equipment LiIon battery failures often result from a short circuit occurring in a cell due to inadequate clearances and either consequent impact sensitivity or hitting the far end of statistical manufacturing tolerance variations. These are design and manufacturing errors that can be avoided at the cost of extra $ - something high volume manufacturers would love to avoid.
In the case of the Boeing Dreamliner battery failures I've not seen a final root-cause report BUT while a number of well publicised failures occurred (and maybe a few unpublicised ones) in a very small product volume, the consequences were astoundingly well contained.
A detailed examination of LiIon failures and modes and consequences shows that they are almost invariably nowhere near as violent as popular 'myth' suggests and that while the energy release is substantial, containment is relatively easy in engineering terms. Containment adds weight and volume and cost and is unlikley to be found in laptops or pocketable / portable devices. It IS found in Dreamliners and could easily be used in automotive single battery (ie non-EV) applications while keeping weight and volume still well below lead-acid levels and at modest extra cost. In electric vehicle applications the problems seem to have been solved or accommodated "well enough". I have ni expertise in vehiclar safety regulatory areas, but am confident that the regulations that bring us spectacular crash-dummy footage and allow the catting of high volatility petroleum fuels in passenger vehicles also address the safety issues around LiIon power sources. I have not heard of a 'Tesla' car being immolated through battery failure - although it may have happened - and I imagine that Musk and co believe they have this risk area "adequately in hand".
I have never, somewhat to my disappointment, seen a LiIon vent-with-flame event and do not personally know anyone who has. Occurrences are common enough to occasionally make the NZ news (NZ population is under 5 million).
LiIon versus LiFePO4:
Compared to LiFePO4, LiIon chemistry offers somewhat better mass and energy densities (somewhat lighter & smaller), substantially LOWER cycle life, slightly lower capital cost (per energy capacity), and substantially inferior whole of life cost. Charging control is about the same but LiFePO4 are significantly harder to damage in marginal cases. Temperature ranges are not as good, charge/discharge efficiency is about the same. LiFePO4 are far less subject to safety issues.
In areas where smallest size and weight and lowest capital cost matter (with electric vehicle use being a good example) LiIon are superior to LiFePO4.
In almost all other areas and applications, LiFePO4 are better or much better than LiIon and I'd consider them the current battery technology of choice for high energy long lifetime, high cycle count energy storage.