If we take some randomly generated key of AES-128 and we change any random 1 byte of that 16 byte key, will this make huge difference in the AES cipher text generated over same input string?
Yes. The outputs with different keys differ greatly. If you pick two random keys the outputs must look completely uncorrelated, or an attacker could gain an advantage just by generating his own key.
With some changes the picture is a bit more complicated, because there are related-key attacks on AES. There the attacker would have to get access to encryptions with keys derived from the original, but if they did they would have an advantage since the outputs aren't totally independent.
However, that's not a very realistic attack in practice, and anyway AFAIK they only exist on AES-192 and AES-256, not (the full 10 rounds of) AES-128.
Does this 1 byte change makes output look like uniformly random? Does it have an effect on security?
In the absence of better attacks, yes. It would be good for security if it was always true, but it's not a huge problem if it isn't, as long as random keys still produce unrelated-looking ciphertext.