There are two complementary systems in our body: Sympathetic nervous system (SNS) and Autonomic nervous system (ASNS). An excitatory neurotransmitter promotes the generation of an action potential in the receiving neuron, essentially "activating" it to pass along a message. Both these neurotransmitters activate different systems so they act differently.
The difference is due to the receptors with which they act. Adrenaline acts on andrenergic receptor. These act on the SNS.
When the SNS system is activated, you get a "fight-or-flight" response. Examples include increases heart activity. This includes increasing the heart rate, conduction velocity, and contractility of the myocardium.
Acetylcholine meanwhile acts on the parasympathetic nervous system. However, it acts on sympathetic nervous system as well, those found innervating the sudoriferous (sweat) glands and some blood vessels of non-apical skin.
When the PSNS system is activated, it leads to effects opposite to that of fight-flight for example, bronchoconstriction, contraction of the detrusor muscle of the bladder, etc.
Sources:
1. ScienceDirect
2. NCBI article