OK,
Let me see if I can lay out the pros and cons of both. I'm a vinyl pool owner so I AM biased.

Vinyl Pros
Expense
Flexibility--if the ground shifts it can move with it.
Fast installation
Easy Replacement
Low maintenance--Brushing and proper water maintenance.
Durability--Expect between 5 and 20 years, but 8 to 15 is most likely
Chemical neutrality--it doesn't affect your water
Easy to add more returns or lights

Vinyl Cons
Appearance
Fragile--it can be cut by tools, doggy claws, some robotic cleaners
Can never be drained until replacement time
Vulnerable to chemicals--excess chlorine fades it, excess acid can soften or melt it. pH MUST be 6.9 or greater.
VERY brittle in winter
Repairs are visible.

Now for tile/plaster/concrete

Pros
NOTHING looks as good as a beautifully tiled pool
With maintenance, can last for decades or longer
Can tolerate FAR higher chlorine levels and far lower pH levels without damage
Durable. Claws, tools, robots can't make it leak
Can be drained dry (with care that ground water isn't high)
Repairs can be invisible

Cons
Expense.
Time of installation can be much longer than vinyl.
A bad installation can be expensive to fix.
A drained pool can float out if ground water is high--far more expensive than replacing a liner.
Repairs including resurfacing frequently are necessary every 5-10 years
New pools affect water balance, leeching calcium and raising pH.
Calcium levels must be maintained between 200 and 400ppm.
Algae can get into grout in a way it can't in vinyl.

Anybody else can add pros and cons to both.