man wget explains:
Don't set the local file's timestamp by the one on the server.
By default, when a file is downloaded, its timestamps are set to
match those from the remote file. This allows the use of
--timestamping on subsequent invocations of wget. However, it is
sometimes useful to base the local file's timestamp on when it was
actually downloaded; for that purpose, the
--no-use-server-timestamps option has been provided.
That is, to make the timestamp on the downloaded file represent when it was created on the local machine, use the
ek@Io:~$ wget https://pbs.twimg.com/media/ByEt7bVIQAAzDVq.jpg --no-use-server-timestamps
--2015-04-05 00:04:55-- https://pbs.twimg.com/media/ByEt7bVIQAAzDVq.jpg
Resolving pbs.twimg.com (pbs.twimg.com)... 18.104.22.168
Connecting to pbs.twimg.com (pbs.twimg.com)|22.214.171.124|:443... connected.
HTTP request sent, awaiting response... 200 OK
Length: 19846 (19K) [image/jpeg]
Saving to: ‘ByEt7bVIQAAzDVq.jpg’
ByEt7bVIQAAzDVq.jpg 100%[=====================>] 19.38K --.-KB/s in 0.04s
2015-04-05 00:04:56 (538 KB/s) - ‘ByEt7bVIQAAzDVq.jpg’ saved [19846/19846]
ek@Io:~$ ls -l ByEt7bVIQAAzDVq.jpg
-rw-rw-r-- 1 ek ek 19846 Apr 5 00:04 ByEt7bVIQAAzDVq.jpg
The advantage of the default behavior, as explained in the manpage, is in case you later want to retrieve the file again, automatically downloading it only if a newer one is available.