$\boxplus$- key addition vs $\oplus$- key addition GOST cipher , Hight cipher and SEA cipher use key addition based on modular arithmetic addition (module key size $\boxplus$) compared to many ciphers which use arithmetic addition modulo 2 ( $\oplus$).






Q.1 what is the difference between two operations in term of security evaluation and performance across software and hardware ? 

Q.2 why do not we see lots of ciphers apply  $\boxplus$ in key addition ? is it because of lack in mathematical proof?
As pointed out in the comments, one attraction of mixing $\oplus$ and $\boxplus$ is that it makes cryptanalysis harder due to non-associativity. 

The differential properties of $\boxplus$ were analysed here and you can find out more by chasing citations to this paper. Massey's PHT (Pseudo Hadamard Transform) is also of interest and was used in SAFER and Twofish.

In terms of key addition using $\boxplus$, the diffusion is slower than XOR when there is no carry, with no added benefits and that is probably why it is not used widely. And the somewhat annoying fact that during decryption, to undo addition of $k$ you must subtract it.