It seems like neolithic is distinguishable by the fact that it involves the development of agriculture and all of the advancements that came with that. Paleolithic is a very long period that encompasses almost everything up until that time... so what is Mesolithic? Is it just sandwiched in between the two because we like things better when they come in threes?
(Sorry for my english) Mesolithic is just the name of the transition period between hunter-gatherism and farming. I don't know about Europe, but in the Levant, think that that change is not a thing of a few years, it's a process that last about 3 millenia (from ~12.5k cal b.C. to ~9.5k cal b.C., Natufian culture), because mesolithic started as soon as climate started to be warmer at the end of the last glacial period and it was easier to get food, making communities more sedentary-like. That process started about two millenia before the start of Holocene (Holocene starts at 9.7k cal b.C.), and, a couple of centuries later, started farming with pre-pottery neolithic A, ~9.5k cal b.C.).
Obviously, every thing happening in that period deserves its own name.
In the "English-tradition" is a bit more complicated because mesolithic is not just used for cultures which later became farmers, but for any changing culture due to the end of the last glacial period (and not the last ice age!! we are still in an ice age!!). However, in Europe we use "Epipaleolithic" for cultures which didn't become farmers (until they did it, but by acculturation), and reserve "Mesolithic" for transition cultures.
NOTE: About dating these periods, it's important to say that, when dating died organism using the radiocarbon method (the method used to date mesolithic and neolithic cultures), we need to know the C-14 environment concentration when that organism died, because we need to know how many C-14 molecules had lost that organism from dead to present.
But, some decades ago, scientist hadn't an accurate measure of C-14 concentrations in the past, and dating was made under the assumption the C-14 environment concentration is the same as present. Today, however, we have a best idea of C-14 concentrations in the past, and dating must be calibrated with that information, to adjust dating.
The thing is, when comparing uncalibrated and calibrated dates from the past, the older the date, the bigger the difference. From mesolithic dates, there's a difference of about two thousand years, and from neolithic dates, of about one thoursand years.
For these reason, if you compare dates from sources of about 10 years ago, you will see that neolithic is dated as started at ~7.000 or ~8.000 b.C., but today is dated as started at ~9.500 b.C.