Battery versus Supercapacitor
I recently read an article ... that seemed to indicate that "supercapacitors" are starting to approach the energy densities of batteries for some applications.


It's possible that there may be niche applications where that is true (although none come to mind with a quick musing) but for even 'ordinary everyday' batteries they have a way to go yet as regards either mass or energy densities.  

BUT, as can be seen below, they have some utterly fantabulous specs that batteries cannot hope to match. eg 1,000,000 cycle life, 1000A+ max discharge current, 100A test current, ... !

Modern high end NimH batteries have energy densities about the same as typical LiIon cells. I'll use a typical NimH AA cell for comparison, but results would be similar for Lion or LiPo. LiFePO4 has perhaps half the energy density of LiIon but even LiFePO4 is far more energy dense that good supercaps. 

An eg AA (14500) NimH cell weighs about 33g and provides about say 2500 mAh at 1.1V mean. That's conservative. Energy = 2.5 Ah x 3600 s/hr x 1.1V = 9900 Joule.
Say 10,000 Joule.  

A capacitor discharged from Vmax to Vmax/4 delivers 15/16 of it's energy (as E= 0.5 x C x V^2).
So discharging a say 2.7V capacitor  to about 0.675V uses most of the stored energy and is still a high enough voltage for operating a boost converter. A boost converter at around 0.6V has lower efficiency than at say >= 1 V but efficiency is liable to be acceptable if accessing stored Joules is more important than maximising efficiency. 

E = 0.5 x C x V^2 x 15/16  = 9900  so
C = E x 2 x 16/15 /V^2 = 2897 F
Say ~= 2500 to 3000 F at 2.7V.  

Digikey cheapest in that range are Maxell K2 series  




  2000 uF = 61mm dia x 102 mm long 360 g  $55/1, $44/250
  3000 uiF = 61mm dia x 138 mm long 510 g $60/1
  AA Nimh = 14mm dia x 50mm long    33 g $3/1 ?  


Cycle life  


  Cap - 1,000,000
  NimH - 500  


Short circuit current - Amp (also abs max for caps)   


  2000 F 1500 A
  3000 F 1900 A
  Nimh ... 10 A  


Toperate C max/min  


  Cap  +65 / -40
  NimH  -45 / 0




Lifetimes:

A 1,000,000 cycle life is quoted but temperature modified calendar life is liable top be the limiting factor. Data sheets for several brands claim 10 year lifetimes at 25 C with the usual Arrhenius equation effect of halving lifetime for each 10 degree C rise in operating temperature. If due care was not taken there are many locations where a 35c operating temperature could occur very easily, with a consequent 5 year typical lifetime.
There will be applications where forced air cooling and even heatsinking may be useful. 



Here's a teardown of a satellite interfaced fishing buoy - Mikes electric stuff August 2014. 

At this point: https://youtu.be/mY2X-ZQpnvY?t=475
You see this. Obviously the cost is irrelevant in the circumstances and the advantages outweigh the fact that this has about the same storage capacity as a good AA Nimh cell. There is space for a second one, but only one is fitted.