It only sounds identical because you're not used to the phenomenon of long consonants/double consonants from your native tongue. But there is a difference--quite a big one, to Italians! There isn't really a PAUSE when pronouncing them, contrary to Huy B's comment, it's just a matter of holding them for longer than you do with short/non-double consonants.
Here's a resource that you might find useful: http://www.italianlanguageguide.com/pronunciation/consonants/double-consonants.asp
Not only is there a double consonant in "cappelli", as already noted, but also the "e" is pronounced differently.
In Italian the letter "e" can be pronounced in two different ways:
- «open "e"» (IPA: /ɛ/, like in the English word "bet")
- «closed "e"» (IPA: /e/, like the 'e' in "grey").
In "cappelli" the 'e' in open; in "capelli" the 'e' is closed.
I cannot see the need to use 'dei' in this sentence as 'Ho cappelli colorati' is what is used according to my friend, a native Italian speaker.
I guess that happens when there are differences between American and English grammar and we're both sets trying to learn a new language and relate it to our own. In America, it is I have, or I have gotten. Never I have got. You would be so marked down on your grammar exercises if you wrote that! :D