Modal analysis tells you the frequency and shape of one of many possible modes that are inherent in the structure. Modal analysis is limited to linear systems, so no nonlinear materials or nonlinear contacts or large deflection effects.
Actual behavior of a real structure might have many modes combined into a complex shape of vibration. Transient Structural analysis shows how a real structure will respond to any general input force or acceleration, periodic or not and can accommodate all nonlinearities.
Harmonic Response analysis shows how a real linear structure will respond to a harmonic (periodic) excitation at a particular place at a particular frequency. If you are very careful, you can excite a real linear structure so that is is almost purely vibrating with Mode 1, but if you move the excitation load to a different place, you will get more modes contributing to the vibration. Harmonic Response cannot have nonlinear materials, nonlinear contacts or large deflection effects.