Now here's the long answer.
Well, if you ask your average public school English teachers, they'll likely tell you that "totally differently" is correct because different is an adjective and you need to use an adverb in this sentence. However, if we take a look at our trusty old sword, we find that "different" is an adjective and an adverb. So both are correct, but some people probably prefer "differently" because of the traditional ly suffix used with adverbs.
different: Definition, Synonyms from Answers.com
In a different way or manner; otherwise: “Carol … didn't know different until Elinor told her” (Ben Brantley).
Let's see what the people say:
"they do things different" - Google Search=
Results 1 - 10 of about 49,200 for "they do things different". (0.07 seconds)
It's by no means close to 126,000, but there's no denying that 49,000 is certainly a substantial number.
"they do things differently" - Google Search
Results 1 - 10 of about 126,000 for "they do things differently". (0.09 seconds)
Another tricky word is "slow". It's both an adjective and an adverb.
slow: Definition, Synonyms from Answers.com
So before you let someone say "Hey, that's an adverb! Use an ly suffix", check with our trusty old sword - the dictionary.
soft - One can speak soft, and one can speak softly.
soft: Definition, Synonyms from Answers.com
In a soft manner; gently.
Granted, most of the time ly is required when it comes to ensuring that one is using an adverb and not an adjective. However, there are certain words that can be used as both an adjective and an adverb. I wish had a list. It's probably a short one.
Do you have any money riding on this question? You might want to place a bet. Just be insistent - and then - bam - show them the dictionary page.
And yet here is another one. One can talk loud, and one can talk loudly.
loud: Definition, Synonyms from Answers.com
adv., louder, loudest.
In a loud manner.
What do the people say?
"talk loud" - Google Search
Results 1 - 10 of about 112,000 for "talk loud". (0.07 seconds)
"talk loudly" - Google Search=
Results 1 - 10 of about 97,300 for "talk loudly". (0.07 seconds)
So that's three so far: different, soft, and loud.
And here's another one. We can say that the snow is falling heavier or the snow is falling more heavily.
adv., -i·er, -i·est.
Heavily: The snow is falling heavier tonight than last night.
That's five now: different, slow, loud, soft, and heavy. All five can be adverbs and you don't need ly. That's great.