For a unique number to represent the sequence, you could consider a hash function. However, hash functions will give you ugly output that won't be readable
and may not be easily reversible. It doesn't seem like that's what you're looking for. If you can't say for sure that your function will have integers, a hash table is a good way to do this. But, the hash values still won't be directly comparable in the way that you seem to want.
For the sequence 1, 2, 3, 4, 3, 2, 1, you could simply consider the unique number 1234321. This might depend on the range of your numbers - it would help/may only be reasonable if you can put some bound on the numbers in your sequence. (if you know all the numbers are integers between 0 and 9, this is fine - we can extend this to numbers between 0 and N (or a and a+N) by representing the sequence in base N+1). This may not save you any space, though - certainly not compared to a hash table.
If you know for sure that your terms will only go up and down by 0 or 1, you could represent them by the string $+++0−−0−$, for example, or equivalently as the number $22201101$ (where 2 is up, 1 is down). As an aside, this could be stored very efficiently by working in base 3. This seems like a good solution for you.
Similarity can be measured in several ways (e.g Hamming distance). In the case of your example, where you have a "deletion" (a 4 has disappeared), it seems sensible to consider the Levenshtein distance. Levenshtein distance applies to strings of text characters, really, but you can easily represent your sequence this way as described in the paragraph above.
It's difficult to understand exactly what you want without more context, but I hope this will help you to research what you need! Further comments welcomed.