Difference between UF-CMA and UF-CMVA security
Why if the tag $\varphi$ is unique the two Games are identical?


If your Tag algorithm is deterministic, then verifying a tag-message-pair is actually just re-computing the tag for the message and checking if the provided and the re-computed tags match. So in this case $\operatorname{Vrfy}_k(m,\tau):=\operatorname{Tag}_k(m)\stackrel?=\tau$. So I hope now you see how step 3 is identitical in both games if $\operatorname{Tag}$ is deterministic.


  But I can't understand why they are different?


Because there are MAC algorithms for which $\operatorname{Tag}$ works non-deterministically, that is with large probability two tags for the same message will differ. Now you can no longer check whether a tag-message-pair is valid as above, you need a more complex verification algorithm.

As a contrived example consider the scheme $M$ which is HMAC with the following modification for the tag-generation: First you generate a random 128-bit string and prepend it to the message, then you run the modified message through the HMAC and return the concatenation of the random string and the HMAC-tag. Verification now works by taking the prepended random string and prepending it to the message before feeding it into the HMAC oracle.