Difference between the inactivation of neurotransmitters and hormones
Short answer
Hormones act in the order of minutes or hours. Neurotransmitters in the order of milliseconds. Moreover, hormones are blood-born, neurotransmitters are confined to the synaptic cleft or in the extracelular space directy surrounding the neuron. Hence, their mechanisms of inactivation are different.

Background
The distinguishing feature of hormones and neurotransmitters is the fact that the former are generally released into the blood, while the latter are confined to synapses, or the immediate extracelullar neuronal space. Hence, hormones can be readily cleared by the workhorse of blood clearance: the liver. For example, adrenaline is rapidly (half life 5 - 10 minutes) metabolized mainly by the liver, followed by excretion via the kidneys. Diffusion then will draw adrenaline away from the receptors as blood concentration drops. The relatively slow time course is beneficial, as the effects of epinephrine includes metabolic changes, such as the release of glucose to power up the body (Sherwin &amp; Saccà, 1984). This needs time.  

In the synapse, there is no blood flow and inactivation has to occur through other means. Further, epinephrine in the above example is a fast acting hormone. Many hormones work on an even slower time frame. For example, a stress-induced cortisol peak can last for as long as an hour (Kirschbaum &amp; Hellhammer, 2000). Neurotransmitter responses may last for a few milliseconds. For example, some pacemaker neurons may fire at rates of 40-50 Hz (Häusser et al., 2004). Hence, neurotransmitters have to be cleared from the synapse at least within 20 ms. To wait for the released neuroransmitter to passively diffuse out of the synapse is simply too time-consuming. Neurotransmitters act at an entirely different time scale.     

References
- Häusser et al., J Neurosci (2004); 24(42): 9215-9
- Kirschbaum &amp; Hellhammer, Encyclopedia of Stress, 3 (2000)
- Sherwin &amp; Saccà, Am J Physiol (1984); 247(2-1): E157-65