I worked on a quantum computer for several years and can speak to this a bit: sorta. They're functionally equivalent in the sense that you can do the same computations, usually, but there are a ton of details that make how each particular modality behave. Things like gate fidelities (how good the gates are), how fast the gates can "execute", how long it takes to initialize the quantum state so you can execute gates, how long decoherence times (how long before the quantum state is lost) are, and many (many) other differences. Some modalities even have restrictions on what qubits can interact with other qubits which will, among other things, impact algorithm design.