They're very different. Requests are random tasks completed, usually, just for Exp. Quests are Story related events. How does this work? It's fairly easy. Requests can be broken down into categories. So far there are only three but there may be more as the site progresses. The Categories are Search and Rescue, Find My Lost Item, Beat That/Those Pokemon. Occasionally you'll get two mixed together in which perhaps a pokemon has stolen an item. Anyway, to take on a Request, note the NPC character in the RP with the Request by their name and speak with them. They'll give you a brief over view and ask if you'd help them. If you say yes, you're officially on a Request. You cannot take on more than two requests at a time and you can't leave the location while on a request unless that request calls for it. Quiting a Request will subtract 1000 Poke from your funds so I suggest you not bite off more than you can chew. Also, if you take on a Request then the whole group goes with you...lovely, right? Standby is a great feature in case more than one person has a request or in case we really need to get moving. Standby is putting a request on hold until later. But if you're mid-way through a request and go for a standby, you'll have to start over later. To go on Standby and move on, you either call Professor Kimba or Officer Jenny. Prof Kimba will send Kirlia to pick you up and drop you off at any location within a one town range. Officer Jenny will show up to relieve you so you can get going. You retake a Quest by going back to that area. An option will appear that says "Resume Standby Quest". Say yes and you'll resume the quest from the beginning.
A Quest is Far Different. A Quest HAS to be done eventually to continue onto the next town etcetera. To take on a Quest, there isn't anything special you have to do. The Option of a quest is always open so all you have to do is follow the quest. EVERY LOCATION has at least 1. Complete a Quest and you unlock the next place, event, etc. A Quest can't be placed on Standby. Once you start one, you have to finish it. Quests, unlike Requests, varry greatly.