I have the exact problem in 13.10 as this user Two Wifi indicators in panel.

Here are some screenshots:

Here are some screenshots from another user: http://ubuntuforums.org/showthread.php?t=2183020&p=12825563

ifconfig and iwconfig outputs

$ ifconfig
lo        Link encap:Local Loopback  
          inet addr:XXXXXX  Mask:XXXXXXX
          inet6 addr: ::1/128 Scope:Host
          UP LOOPBACK RUNNING  MTU:65536  Metric:1
          RX packets:2243 errors:0 dropped:0 overruns:0 frame:0
          TX packets:2243 errors:0 dropped:0 overruns:0 carrier:0
          collisions:0 txqueuelen:0 
          RX bytes:209889 (209.8 KB)  TX bytes:209889 (209.8 KB)

wlan0     Link encap:Ethernet  HWaddr XXXXXXXXX  
          inet addr:XXXXXX  Bcast:XXXXXXXX  Mask:XXXXXXX
          inet6 addr: XXXXXXX Scope:Link
          UP BROADCAST RUNNING MULTICAST  MTU:1500  Metric:1
          RX packets:5925 errors:0 dropped:0 overruns:0 frame:0
          TX packets:3361 errors:0 dropped:0 overruns:0 carrier:0
          collisions:0 txqueuelen:1000 
          RX bytes:2951818 (2.9 MB)  TX bytes:630579 (630.5 KB)

$ iwconfig
lo        no wireless extensions.

wlan0     IEEE 802.11abgn  ESSID:"XXXXX"  
          Mode:Managed  Frequency:2.437 GHz  Access Point: XXXXXXXX
          Bit Rate=72.2 Mb/s   Tx-Power=15 dBm   
          Retry  long limit:7   RTS thr:off   Fragment thr:off
          Power Management:on
          Link Quality=49/70  Signal level=-61 dBm  
          Rx invalid nwid:0  Rx invalid crypt:0  Rx invalid frag:0
          Tx excessive retries:153  Invalid misc:472   Missed beacon:0