Open your network preferences and force wireless to the top of the list. If wireless is not top of the list when ethernet is connected.. just click on the options button at the bottom and click on set service order.
Change the service order so wifi is always on top if there is a wifi connection.
The next bit is more complicated.
Just doing the above should fix it but it can still fail if the IP address and particularly gateway address is wrong. (To add confusion apple use router for gateway.. but I will stick to the correct term).
Your ethernet needs to be on a different IP range to the wireless connection.
So if you have a typical setup lets say your wireless is 192.168.x.x
Then your TC connection is at default 10.0.1.x with gateway set to 10.0.1.1 then that is fine..
You do not want both at 10.x.x.x (even with correct subnet).
But you can use 192.168.1.x for internet via wifi and 192.168.2.x for TC for example.
Tell me or give me screen shots of the wifi IP and the TC IP if you are confused and I can tell you how to fix it.
What is wrong.. is you now have two Gateways (router in applespeak) and two DNS servers .. one of which is wrong. The gateway and dns on the TC go nowhere since it is isolated network.
The solution is to fix the IP manually in the computer.
Lets say you are using TC on default.. and wireless on 192.168.x.x which is easiest.
Set any address you like on 10.0.1.x but it is easiest to use 10 or 100.
You do not need a gateway.. (router) but because I discovered it is unhappy being empty.. stick the same Router address as the wifi has.
In my example.. 192.168.1.1 but this has to be correct.. and same as wifi.
You are nearly done.
You also need to fix that wrong DNS address.
So go to the advanced and DNS tab. Click the + at the bottom to change it.
You could leave it blank but again you are better filling it in with the same value as the wifi connection.
Say 192.168.1.1 or your ISP values.. or google DNS.. 8.8.8.8
Then your TC is part of the local network.. but has no ability to take control of the computer routing.. gateway and dns are correct.
I hope that all makes sense..
The only other comments
N.B.
1. This is a long way around to a better solution.. plug a USB drive into your computer and use that.
Even USB 2 is faster than a TC over gigabit.. it is more reliable.. especially cf a Gen3 TC.. it works better on local drives than network drives for Time Machine.. and it is cheap and easy to get 2TB or larger drives.
2. Never forget you have set statically if you change your network around.. you will have ethernet with the wrong settings and will not get any internet. You need to go back to auto settings to wipe out the changes you made.. it is something people do forget and then they wonder what has gone wrong.
