As indicated by narx's answer to a similar question on ELU (English Language & Usage, ELL's "sister site" aimed primarily at linguists, etymologists, and serious English language enthusiasts), only little can be used as an unqualified noun to mean a small amount (where adjectival small needs an explicit noun, such as amount or quantity).
The other interesting point made about little there is that in...
1: I have a little experience teaching English.
2: I have little experience teaching English.
...the two sentences have almost opposite meanings - #1 implies you have at least some experience, whereas #2 implies very little (perhaps almost none). In fact, #1 is often used somewhat facetiously to imply you actually have a lot of whatever is being spoken of.
Apart from the above distinctions, there's also (as kiamlaluno points out) the fact that little often carries more "affectionate" connotations than small. And from BBC Learning English, little is more common in the sense not much when coupled with abstract nouns such as hope, chance, change, effect, use and point.
In most other contexts, there's little to choose between the two words. Use whichever you like.
'Few' is mainly used when talking about the number of 'count nouns', such as 'dogs'. You can say
Few dogs make good friends with cats!
to mean that not many dogs make good friends with cats. But if you said
Little dogs make good friends with cats!
this would suggest that dogs that are small (cat-sized, perhaps?) make good friends with cats.
On the other hand, 'little' is used when talking about the amount of 'mass nouns', such as 'water'.
There was a flood, but little water actually got into the house.
means that not a lot of water got into the house (thank goodness!). On the other hand,
There was a flood, but few water actually got into the house.
is not a valid sentence.
Sometimes you can use either to produce equivalent sentences, but usually 'few' will precede a plural noun and 'little' will proceed a singular (mass) noun. For example:
John had little reason to dislike Jim.
Jim had few reasons to dislike John.
Both of these sentences mean that they haven't given the other much cause to dislike them.
A little and a few both mean some, but the above rules still apply here. So you would say a few dogs and a little water, but not a little dogs or a few water (which aren't valid sentences).
Best Answer
There is a difference in meaning between “a little” and “little”.
The meaning of “a little” is positive. It means some or a small amount, such as, “I have a little money.”, “He made a little progress.”, etc.
On the other hand, “little” has a negative meaning. It means an extremely small amount or an amount that is less than expected or wished for, as in, “I have little money.”, “He made little progress.”, etc.
If we look at these sentences, the difference between “a little” and “little” will come across easily. The former may be satisfactory for a particular purpose while the latter is not.