The engagement model you choose shapes incentives for the entire project, not just how the invoice is calculated.
Fixed price
You get budget certainty in exchange for a rigid scope. This model works well when requirements are genuinely stable and well-documented — and works badly the moment requirements need to evolve, since every change becomes a negotiation.
Time & material
You pay for actual effort, which gives you the flexibility to adjust priorities as you learn. The trade-off is that budget discipline shifts to you: without active involvement, scope and cost can both drift.
A simple rule of thumb
Choose fixed price when you're commissioning a well-specified, bounded piece of work from a vendor you don't yet trust deeply. Choose time & material when you're building a product that will keep evolving with an established, trusted team.