Fellowships are personal awards to an individual, whilst grants tend to be funding a project that involves a group of academics forming a research team.
Fellowships fund the salary of the applicant, while a grant usually allows hiring people to deliver the project. In most cases this means that the applicant of a grant needs to have a salary supported by university, and the grant will support only a small part (e.g. 10-20%) of the salary of the applicant – the time that is calculated as needed to deliver the proposed project. There are exceptions e.g. BHF early career grants.
Apart from funding the salary of the applicant, certain fellowships can also fund additional staff costs.