White sugar vs refined sugar
These are technical terms that are meaningful only in the original Thai. The English translations are meaningless and have no relevance to cooking.
http://www.ratchakitcha.soc.go.th/DATA/PDF/2552/E/132/15.PDF
So you should say 'น้ำตาลทรายขาว' ('white sugar') and 'น้ำตาลทรายขาวบรสิทธุ' ('refined sugar')
น้ำตาลทรายขาว is further divided into 3 classes.
This results in 4 grades:

น้ำตาลทรายขาว grade 3 99.0% sucrose by ISS standard, 0.2% max reducing sugars
น้ำตาลทรายขาว grade 2 99.5%, 0.1% max reducing sugars
น้ำตาลทรายขาว grade 1 99.5%, 0.1% max reducing sugars
น้ำตาลทรายขาวบรสิทธุ  99.8%, 0.04% max reducing sugars

Max humidity and conductivity ash each then then 0.1% for น้ำตาลทรายขาว, and 0.04% for น้ำตาลทรายขาวบรสิทธุ.
These numbers are almost certainly useless to the cook.
The FAO has more useful standard names:
http://www.fao.org/input/download/standards/338/CXS_212e_u.pdf

white sugar - 99.7% pure sucrose
Plantation or mill white sugar - 99.5% pure sucrose
icing sugar - white sugar which has been finely powdered
soft white sugar - Fine grain purified moist sugar, white in colour with a
sucrose plus invert sugar content of not less than 97.0%
Soft brown sugar Fine grain purified moist sugar, light to dark brown in
colour with a sucrose plus invert sugar content of not less
than 88.0% m/m
Raw cane sugar Partially purified sucrose, which is crystallised from
partially purified cane juice, without further purification,
but which does not preclude centrifugation or drying, and
which is characterised by sucrose crystals covered with a
film of cane molasses.

Both น้ำตาลทรายขาวบรสิทธุ and น้ำตาลทรายขาว are in fact highly refined.
Tesco Lotus charges 22THB for น้ำตาลทรายขาวบรสิทธุ and 21THB for น้ำตาลทรายขาว
However if you refer to

น้ำตาลทรายขาวบรสิทธุ this is 22THB/kg
and this is 34THB/kg

I believe that legally this is also  'น้ำตาลทรายขาวบรสิทธุ' under Thai classification
Caster sugar may also be referred to as 'superfine' sugar. This is quite different from powdered/icing sugar as it is a process of making small crystals, not of grinding sugar to a powder. The smaller crystal size makes it very good for baking as it blends with butter etc. more easily than larger crystals.
Highly refined sugars as all those mentioned here are not necessarily good in that there are lots of flavours in sugar cane (all Thai sugar is obviously cane sugar), and personally I prefer to make for example banana bread (which is cake, not bread) using completely unrefined aren sugar made from arenga pinnata, which is around 0.5% ash, typically slightly fermented, and contains numerous impurities. The taste of the cake is far better than using highly refined sucrose. I'd probably use dark brown sugar if it were cheaper, but where I live aren is the cheapest so I use that.
In Thailand, Indonesia and probably quite a few other neighbouring countries sugar prices are highly regulated, and the terms น้ำตาลทรายขาว and น้ำตาลทรายขาวบรสิทธุ are likely to reflect political issues both as a key grower and exporter, but also consumer prices. Unregulated sugars for rich consumers could be much more expensive.
It does follow that น้ำตาลทรายขาวบรสิทธุ is slightly higher quality than น้ำตาลทรายขาว, in that the goal of both is simply to be pure sucrose (which actually varies slightly because cane sugar and beet sugar, due to different carbon isotopes), and if you were to choose only one, you might as well have the one with the smallest crystals and highest purity (in that order). But the purity is much less important than crystal size.