I see that the question has been answered for SETS in "The Decomposition VS. The Partition of a set" but I would like good definitions that distinguish between these terms when used for NUMBERS. I see both terms used on the net by different people, but they seem to use them in the same way, that is to write a number as a sum. For example 25 = 20 + 5 and 10 = 5 + 3 + 2.
In Number Theory, a partition of a positive integer is an expression of that integer as a sum of (one or more) positive integers of non-increasing size. Thus, for example, $4$ has $5$ partitions; $4,3+1,2+2,2+1+1,1+1+1+1$.
Decomposition of a positive integer, I agree with Joonas, has no canonical definition; you may use it as you wish, provided you make it clear just exactly how you are using it. One common use is the decomposition of a positive integer into primes; this is a decomposition into a product, rather than a sum, as, e.g., $12$ is decomposed into $2\times2\times3$.