It depends on the context, but generally speaking, החל is higher register than התחיל
החל is less common and more poetic.
Having said that, in commercials you can often hear הספירה לאחור החלה (the countdown has begun).
Another difference is that החל is usually standalone. If you want to say "began to" you would say התחיל ל. Almost never החל ל.
Searching Google, I find plenty of uses of the future (I searched for "יחל" and "יחלו"). The present tense is harder to search for, because it has the same spelling as more common words, but if you search for specific phrases like "מחילים לבנות" and "מחילים אחרי", you do get a few results.