Different behaviour of FTDI231XS on two PCBs with same circuit
As others have pointed it already out, I didn't follow the design guidelines for USB connections.
They can be found for example here
or here
For USB connections the informations on page 14 of the FTDI design guidelines are especially important:


Equal length: Both DP and DM signals must travel the same distance. If one trace ends up longer, then the timing of the signals can be adversely affected and cause data errors.
Controlled impedance: The impedance of the twisted pair cabling must be matched on the PCB in order to minimize signal reflections. USB signals are 90Ω differential to each other / 45Ω each to Signal Ground. Most modern PCB layout software can be configured to route both of these signals together with these characteristics.
No stubs: When adding components such as transient voltage protection or additional capacitance for edge rate control, the DP and DM signals should not have any “T”s in order to minimize signal reflections.
Ground planes: With DP and DM being controlled impedance, they should consistently run over the USB Signal Ground plane. There should not be any splits in the plane directly under DP and DM.
Overall length: The DP and DM signals should be made as short as possible. For very short runs, less than 1cm, it may not be possible to observe the controlled impedance specification. In practice, this is usually acceptable provided the other practices are followed.
General design practices: Keep noisy sources away from the USB signals; avoid right angles; etc.



As it has been pointed out by Christian B. the problem lies with a high probability in the different length of the USB D+ an D- data lines. Also the arrangement of the corresponding condensators could be improved.