The buses are seen as distinct by the fare reader. I have never been double charged for transferring from a 1T to a 1W (Scenario: I missed the 13, so I took the 1T to Route 9 and 119 and dashed across the street for the 1W going East )
The branches of the 1 have distinct colors and destinations, but many transit apps and sites do not take note of this. Which bus you take depends on where you want to go. You should decide before you board.
Route 1 only goes to the Yonkers City Line (or in some cases, Mercy College)
In the evenings, the 1 only goes to the City Line. You could also take the 6 to that same stop.
Route 1W goes to White Plains and turns right at NY-119 when Route 1T continues straight.
If you need the Harlem Line, connections to other state systems, or County Offices, you should take the 1W.
If you need the Hudson Line, or connections to Ossining, Peekskill or Briarcliff Manor, you should take the 1T. The 1W will also connect to the 13 and 14.
Route 1C - it depends.
If the wording is Via WMC, then it stops at the college last.
If it says via WCC, it stops at WMC last.
Route 1X is a skip-stop route and utilizes highways and US-9 (it makes one stop along Ashburton Avenue). The 3 goes to Manhattan Road along NY-119, NOT the bus terminal. If you need the terminal, you should take the 1W. There's a shopping mall just beyond the curb.