Ratings

The following is described in the uPD703107 Data Sheet:

There is no clock difference between them. Is this correct?

This is not a mistake.

[36] is the timing at which the output buffer starts outputting and [37] is the timing at which the output data is determined.
Therefore, the area shown from [36] to [37] is indicated by multiple lines and this shows that the data is undefined.

[36] is the specification used to check for bus floating after the previous data read and collision of the bus output with the next data write.
Regarding the write operation itself, use the specification of [37].

Both CLKOUT synchronization and CLKOUT asynchronization are listed as external bus specifications in the data sheet, but is it necessary to satisfy both?

No, satisfying either one of them is fine.

There are two sets of external bus specifications, CLKOUT synchronization and CLKOUT asynchronization, but how are they respectively used?

If only a semiconductor device such as SRAM that can be satisfactorily controlled with address and data and simple control signals such as RD or WR is connected to the external bus, use the CLKOUT asynchronization specifications. If the CLKOUT asynchronization specifications are used, the design of timing with SRAM can be done comparatively simply.
If the device to be connected to the external bus cannot be controlled just by the control signals output from the CPU and more complex control signals are required (when control involving fine timing is required), the control signals may be generated through timing synchronized with the CLKOUT signal. In this case, use the CLKOUT synchronization specifications.

I want to connect an SRAM to the external bus, but due to the speed factor, it is necessary that waits are inserted. In this case, since control of WAIT input is required, must I use the CLKOUT synchronization specifications?

By using the programmable wait function, the bus cycle can be delayed up to 7 clocks. If this range of wait is required, CLKOUT asynchronization specifications can be used, allowing easy bus timing design.