Yes. In formal and technical English, the difference is a matter of precision. "Half the time" is a precise and often a mathematical descriptor. But this is in formal and official English. In informal contexts, the thinking is looser, so it is as rough and crude in reference to frequency so as to only be vague, situationally relative occurrence (based on life context and common sense expectations). Then it means what "often" means. "I often have sleepless nights" can mean 5-10% of the time! "Half the time" never means that little if one thinks carefully and in detail. Anyway, one is mathematical and the other is contextually relative in its base form. "Half the time I go to Walmart" means a reference to 50% incidence in formal English. But in the informal English that is used with family, friends, and coworkers, it is meant to mean often. It's just as vague and imprecise a frequency as "often" is, which means of notable frequency or incidence according to the situation being mentioned or in the rating opinion of the person speaking. "Often" can mean "rarely" to the average person, but frequent to the speaker or writer. "I often eat sushi" means I eat the dish regularly or in a frequency significant to me. And the same is true of many other assessments, ratings, opinions, and qualifications used in casual speech and writing. That is, words of precision like percentages or ratios are used loosely and seen uncritically to the naked eye, so they are taken less seriously and to be less precise. So "half the time" is a loose, inexact, relaxed, almost flippant assessment in casual English or when spoken to your friends, but not when it appears in formal presentations, medicine, engineering, or science. But the two descriptions are taken uncritically; so much so as to be close enough to be synonyms or mean the same thing when said in informal English. They are "close enough" in casual and uncritical references, but not in formal language or writing.