We see those terms more often now that salons offer Brazilian Blowout or Keratin Blowout.
Blow dry is simply drying your hair. You can easily do it no need of a professional hair stylist to do this. You simply get a blower and just move it around your hair until it dries out. This will only take 5-10 mins depending on the length and thickness of your hair. In a blow dry you simply dry your hair.
Blowout is done professionally. This will leave your hair glossy, straight, shinny and full of volume. This will take about 20-40 minutes depending on the length and thickness of your hair. Hairstylist usually dry 50% of your hair, then section it, and do a per section blow dry using a round brush. This needs a pretty good upper arm muscle. This is just temporary once your hair gets wet or you take a shower the effect of the blowout will be gone. Usually clients go to the salon to get temporary blowout if they have occasions to go to.
Hopefully I shared some light on this matter by simply defining the terms used in the salon. The next time you go to a salon you know the correct term to use.