You are all so wonderful. I love this site. First off thank you all for your advice and opinions. Sorry it's taken me all day to respond back.
To answer most of your questions:
I don't yet have a deposit from them, but that was more a delay on my part. The "back and forth" I mentioned was getting them to answer additional questions I had on the credit check that came back, as well as better contacts at their prior apartment complexes. I was not in negotiations about the lease contract. Mine is solid (reviewed and approved by 3 different lawyers) and I have no intentions of changing it.
I've never had to deal with tenants who were prior renters (my other tenants were home owners prior to renting from me) so checking on apartments I've found is cumbersome and time consuming!
All of their information checks out and they seem like solid tenants.
As for why this other potential tenant is willing to offer me more $$, I suppose I could be slightly under market on price, but do feel like I've done decent homework to price the property correctly. It's in Thornton, CO and it's a 2/2 @ 1296sf. I am getting $1,650 for it which honestly at first seemed on the high end, but I've gotten 45 calls/emails on it so you all might be right. Might be under priced.
My non-written policy, but will be written now going forward (thank you @Nathan Gesner) has always been first come first served that meets my requirements.
I also haven't taken the listing down as some of you advised, since I don't have the lease signed and deposit in hand. I plan on finishing up / getting the deposit in the next 24 hours.
I was excited about the extra $200/month, but I think sticking with Tenant#1 is best and I just do a better job scrutinizing my rent price -- would love you advice on how you do that / where you go for 'rent comps'.
Take care!
Angela