Either invisalign or braces can fix your open bite. What treatment is best depends on you. Cosmetically invisalign will look better but you will have attachments on your front teeth. Braces should cost less, probably around a $1000 less.
Open bite an treatment options
the problem in your case is that the most likely CAUSE of your speech problems AND orthodontic problems is your tongue...fixing the orthodontic problems does not guarantee that your speech problems will get any better. Open bites are very hard to correct...make sure you get a second opinion and get treated by an orthodontist