So now it's: www.infoaboutbooking.com and www.bookingengine.com ?
and you want: bookingengine.infoaboutbooking.com ?
The only real argument I can think of is speed. That booking engine has to be very very fast. So it needs his own server. (Free from the webserver that hosts the main domain) Usually something that is hosted on another server is hosted on another domain or subdomain (although you can host 1 domain on multiple servers...)
Using a subdomain vs a different domain then the main domain is a non-discussion. Because in SE's eye's they are both different websites. So it doesn't really matter. Especially because there isn't any content on the booking engine website. (I assume) It's just booking. So think only about your users. Make it very fast and very userfriendly and let the tecchies figure out how to measure stuff between the (sub)domains. (conversion, turnover, etc.)

