Roku made a change in behavior awhile back to their "Continue Watching" with the mobile app. (the section is labeled Continue Watching on TV)
If you have a Device selected under the Devices tab, it will automatically be launched on the selected device/TV).
There are two ways to play on your mobile. If you really want to "Continue Watching" (but on your mobile), go to the "Devices" tab and select "Forget this device" for the device that the mobile app is connected to. (use the three little dots in the upper right of the connected device). Now when you go to Continue Watching on TV, you will be presented an option to "watch on mobile" at the end of the list of available devices.
Other way to watch on mobile is to search for the content (by title) you want and then select the "watch on mobile" option.
Essentially, just make sure you do not have the app connected to a device (by using the 'forget' option) if you want to watch the content on your mobile rather than your TV.
