The nonlinear dynamic behavior of a direct frequency-modulated diode laser with strong optical feedback is examined and compared to a laser diode subject to electro-optically modulated, strong optical feedback. Direct modulation is achieved by sinusoidal modulation of the diode laser injection current. Electro-optic modulation is achieved by applying a sinusoidal voltage to an intracavity phase modulating element. The output state (characterized by the output power versus time, the intensity noise spectrum and the optical frequency spectrum) for both types of modulation is dependent on the ratio of the modulation frequency to the external cavity resonant frequency, and the modulation power. A number of distinct states are observed: conventional amplitude modulation (with FM spectra); multimode, low-noise amplitude modulation; multimode, high-noise amplitude modulation; periodic limit-cycle operation; quasi-periodicity; chaos; low-frequency fluctuations; and mode-locking. There are significant differences between the direct and electro-optic frequency-modulation cases. The onset of the dynamic instability is characterized as a noisy period-one oscillation for direct modulation and a low-frequency fluctuation for intracavity electro-optic modulation. Phase portraits produced experimentally with the use of a digital phosphor oscilloscope are shown to agree well with those constructed from output power versus time data. This represents an experimental method for examining the dynamics phase portraits in real-time.