Using "modern" batteries with old devices
Current (Amps) is not the total power storage of a battery. Cell phone batteries are all 3.7V so they are often compared by the mAh rating of the battery. In reality to calculate run time you need to use Watt Hour (Wh) which is the total amount of energy stored. 

To find total Energy storage of a battery (assuming battery capacity given in amp hours):
\$E[WattHours] = P[watts]*time[hours] =&gt; V*I*1hr \$.

Old Battery: 7.4v x 2500mAh = 18.5 Wh

Cell Phone: 3.7v x 4500mAh = 16.7 Wh

So your modern cell phone battery actually still holds less power then your "old" Lithium Ion battery in the laptop. While battery technology is always getting better, it is not moving as fast as you may think.

Per my comment, laptops are very power hungry compared to a phone that spends 90% of its time sleeping in your pocket. Also modern electronics constantly require less power to run, thus resulting in a longer run time for a given battery size.

Update About Size:

I missed the 1/4 the size in the question originally. Given that info, It would seem that a modern battery would have a higher power density and thus provide a longer run time. The hardest part would be getting cells that match the batteries form factor, since most laptop batteries are made to fit the form of the unit. If the battery could be created out of typical 3.7V cells that are available on various online marketplaces, then there is a chance you could do something. Although you would still need to ensure the battery has a similar interface for charging and add some sort of a battery protection circuit that basically all multi cell packs have.