In my opinion, neither previous nor prior work in this context for what you want to express, because both expressions imply December 2009.
Also, 'between' is a bit vague, because while the context implies you are comparing the rates of the two months, 'between' carries the idea of comparing the rates for the whole period between those two months.
Why not simply 'Comparison of rates: Jan 2010 and Jan 2009'?
It is not especially clever, but it is shorter than the two phrases you propose.
english.stackexchange.com/users/529/Benjol
Neither "previous" nor "prior" are correct here, because they would refer to December 2009.
If you want to use a business analysis buzzword, you could use "year over year", as in Year-over-year comparison of rates for January If you are addressing normal people, however, I'd suggest simply Comparison of rates between January 2010 and January 2009
english.stackexchange.com/users/21/Peter Eisentraut
The Oxford Corpus of English lists several example sentences for prior :
She visited me on the day prior to her death.
Mrs Scott had to spend two weeks in hospital, including a week prior to the Caesarian birth.
For previous : The month previous to publication . . .
The starting grids are based on a reversal of the finishing order of the previous race.
Both prior and previous have similar definitions: 'existing or occurring before in time or order' and 'existing or coming before in time, order, or importance' respectively.
From these two points (examples of usage in English and dictionary definition), I can see no evidence that either word is more or less appropriate in your context.
However, I would say the following sound better: Comparison of rate between January 2010 and the prior month.
Comparison of rate between January 2010 and the previous month.
That is, I would use the rather than its , although your sentences are grammatical.
EDIT: Now that I understand that the OP was talking about comparing January 2009 and January 2010, I offer these suggestions: Comparing the rates of January 2010 and the prior January.
Comparing the rates of January 2010 and the previous January.
I also agree that it would be easier, and perhaps better, just to say 'Comparing the rates of January 2009 and January 2010.
*
english.stackexchange.com/users/1065/J D OConal
In general for comparing corresponding periods across years you use the expression year on year
english.stackexchange.com/users/122/cindi
Neither prior nor previous are words necessary to convey the meaning of the statement.
The clearest way, so that no one misunderstands the question, is to delineate it completely: Comparison of rates between Jan 2009 and Jan 2010.
This "statement" implies that there is a list to follow with the said comparisons indicated.
It is not meant to convey all of the actual comparisons.
It seems to be more of a heading.
english.stackexchange.com/users/12559/Franc Mosbaugh
If I want to show the comparison between rate in 2010 Jan and 2009 Jan, which of the following should I use?    Comparison of rate between 2010 Jan and its prior month.  Comparison of rate between 2010 Jan and its previous month.
Can "prior" or "previous" be used to describe the same month of last year?
english.stackexchange.com
<word-choice><phrases><adjectives>
english.stackexchange.com/questions/3893
english.stackexchange.com/users/748/lokheart
It is agreed that neither "prior" nor "previous" work in this context because they would refer to December 2009. Neither "previous" nor "prior" are appropriate because both imply December 2009. It would be simple to just say "Comparison of rates: Jan 2010 and Jan 2009".  The other alternative is to use the business term "year on year" . Another alternative is "comparing the rates of January 2010 and the previous January's". 
It is agreed that neither "prior" nor "previous" work in this context because they would refer to December 2009. Neither  prior  nor  previous  are correct here for conveying the meaning of the statement.  The clearest way, so that no one misunderstands the question, is to delineate it completely:   Comparison of rates between Jan 2009 and Jan 2010.   
 Neither "previous" nor "prior" are correct here, because they would refer to December 2009.  It is agreed that neither "prior" nor "previous" in the examples provided is correct.
Neither previous nor prior are suitable. Use "compariso of rates" Use "year on year" Use "comparing the rates of Jan. 2010 and the previous January"
Neither  prior  nor  previous  are correct here for conveying the meaning of the statement.   The clearest way, so that no one misunderstands the question, is to delineate it completely:   Comparison of rates between Jan 2009 and Jan 2010.   
It is agreed that neither "prior" nor "previous" in the examples provided is correct.