There's no difference in meaning, but there is a significant (though subtle) difference in usage and emerging trends. With verbs relating to actions, both to start to [act] and to begin to [act] have long been about equally common...

But for verbs concerned with internal mental states, desires, etc. we nearly always use began to...

If you check this NGram for began to / started to feel, you'll see that the latter version has at least started to gain some traction. But that doesn't invalidate the broad principle. Besides which, at least some of the instances contributing to that chart would be for literal / tactile feeling, not an emotional state.