In a model with specified trend, e.g. technology shock A_t is composed of a permanent component A_{t}^{p} and a transitory component \varepsilon_t, and A_{t}^{p} has A_{t}^{p}=A_{t-1}^{p}\varphi _{t}. What’s the economic difference between \varepsilon_t and \varphi _{t}? And could I just neglect the transitory component \varepsilon_t and only consider about the permanent component \varphi _{t}?

Permanent and transitory shocks have different effects. For the transitory one, agents know that it will ultimately vanish, allowing them to smooth the time path. That is not possible for permanent shocks.

If I do not have enough observed variables, could I neglect the transitory component \varepsilon_t and only consider about the permanent component \varphi_t?