I have worked in organisations that rely heavily on 3rd party cloud servers.
In a small organisations you are given external IT support and can reduce the costs of having in-house servers.
Flexibility in expanding or reducing IT infrastructure requirements (assuming that your contract is not too restricted).
Upgrades handled by 3rd party.
Dependence on external network
Dependence on response times of 3rd party. Service Level Agreements need to be agreed with cloud supplier.
Infrastructure performance can be impacted by excessive use by other customers of 3rd party cloud. Good 3rd party cloud suppliers should ensure this doesn't happen.