I am mainly talking about QSVM from Qiskit (https://qiskit.org/documentation/stubs/qiskit.aqua.algorithms.QSVM.html#qiskit.aqua.algorithms.QSVM) versus a classical SVM. Is it just a time complexity speed-up?
The main benefit is quantum computing may enable the use of kernels which are hard to compute classically. In other words, it may be possible to separate the input data using quantum feature maps which are not available in classical calculations.
You may find this paper helpful - it discusses the issue in some detail.