It is a BN81 1725/3250 with L1, L2, A, B white and brown wire connections. I get it to run on white on L2, Black on B, but I cannot get low speed to run. What is correct connection for my white, red, and black wires to go to my controller? Thanks

So you don't need the Brown?
Couldn't find a BN 81, did find a BN 91:A.O.Smith Electrical Products Company

I might need brown. Does it go to my red wire from controller?

Can't say,
Look at the Name plate REAL close, or the connection cover for the wire descriptions, I had no luck on their site with your number, For your controller it may only need to use 2 speeds, Need to figure connections still for correct voltage and direction as well.
Look close at name plate. And/Or verify Model Number.

Pump Motor label reads
Cat BN-81 Ser. BX 1-28
Wiring shows follows
Low Voltage
Brown-B
White L2

High Voltage
White-B
Brown-A

Pump Duty is 1081/1563
115/230Volt

My controller wires to attach are green, white, black and red.

Does it show any lines or connections between your low and high voltage, its only telling part.
You show 2 wires for low and 2 wires for high voltage, but need more info for the 2 speeds.

Nothing between the two but a thin vertical line sepearting each wiring diagram. It reads as follows:
Low Voltage
Brown-*B
oL1-Line
White-*L2-Line
0A

High Voltage
White-*B
0L1-Line
oL2-Line
Brown-*A
*= Filled in circle
Non-Reversible

Found your Motor, it APPEARS to be single speed. Dual Voltage.:
AO Smith Thru-Bolt Mount Motor - Century - 1 HP - 230/115 Volts - 48Y Frame - Stock Number BN81

Can this pump run on the same 115Volt 20 amp circuit as the spa electric heater? Or will it trip the circuit breaker?

May work, need to see what the draw is. Still 20 amp protection is needed.
Is the Heater 240 Volts? Not that it matters.
Most heaters draw a lot.

Heater is 115 volt with thermostat built in, 1.5 kilowatt rated