Which is a better battery for a smartphone Li-Ion or Li-Po?

Let me try to clear up some confusion here.  In the current battery market, Li-ion is the same as Li-Po (or is also called Li-Poly or Li-Polymer or pouch cells); only the packaging material and shapes are different.  In smartphones, flat pouch Li-ion cells are used, because it can be made the thinnest.

LITHIUM-ION 101

Li-ion is a general term for at least 6-different major lithium-battery chemistries that are used for different applications (LCO, LMO, NMC, LFP, LTO, LNA).  Maybe in LITHIUM-ION 102 I'll go into more details about each type.  For now, it's sufficient to know that most cell phones use the LCO chemistry (LiCoO2); because it offers the highest Wh/kg or specific-energy (some calls it gravimetric energy density).

Definitions

Cell = most basic battery configuration, lowest voltage, 1-positive and 1-negative electrode; for a LCO pouch cell, it is 3.7V nominally.

Pack = multiple cells connected together to yield a higher voltage or higher current battery.

Energy density = energy/volume (Joules/cc, J/liter, Wh/cu-ft, Wh/liter, etc.)

Volumetric energy density = same as above (Wh/liter most popular)

Gravimetric energy density = energy/weight (J/g, Wh/kg, etc.)

Specific-energy = same as above (Wh/kg most popular)

Specific-power = power/weight (W/kg)

C-rating = ratio of charging or discharging current from a cell or pack compared to its Ah rating.  For example: a 6Ah cell means it can discharge 6-amps for 1-hr.  So, 1C means it is discharging at 6A; or 2C means it's discharging at 12A, etc.  Typical charging C-rating of 0.5C means it is being charged at 3A.  RC airplae batteries are formulated differently, in order to provide the high C-rating capability (> 5C) without catastrophic degradation.  All Li-ion chemistries are capable of high C-rate discharge, and all will experience accelerated degradation with fire and explosions in extreme cases (except LTO, my favorite Li-ion chemistry).  LFP heats up less than LCO, LMO, NMC and LNA from it phosphate component.

Shapes

There are 3-basic configurations or shapes for Li-ion battery cells.

1. Flat pouch cells - uses aluminized polymer film heat-sealed with flat foil terminal tabs for each electrode.  Cell phones and many other applications requiring either thin profiles or max packing-density in a limited space use pouch cells.  Many digital cameras and camcorders has plastic case around the pouch cell, because these batteries are frequently removed for charging or replacement.

1. Cylindrical cells - the two electrodes are stacked together (w/separator film), and then rolled into a cylinder.  The advantage of this design is a metal casing tightly wrapping around the cell to minimize expansion of the cell as it warms up during discharging/charging; which helps to maintain good performance and increases cycle-life.  The most common round cell is the 18650, then the 26650.  18 or 26 means 18 or 26 mm in outer diameter, and 650 means 65 mm in length.

1. Prismatic cells - essentially a rolled-up cell put into a rectangular metal or plastic casing with either post or threaded terminals.  You can roll things up in elongated shapes, so not always have to be circular.  These cells are mostly used for electric buses, trucks, solar/wind storage for their high current and high kWh capabilities.

Original Lithium-Ion Polymer Battery - It was patented by Bell Labs back around the mid-1970's, and it used a solid polymer electrolyte (gel qualifies).  It can be made into 3D shapes.  I even did a conceptual design of a backpack with an internal frame using this new Li-Polymer battery.  It was never commercialized due to lower specific-energy and a lower voltage; very unfortunate.

I'm guessing the people who coined Li-Poly after seeing the pouch cell, never knew about the history of the real Li-ion Polymer battery.