Sometimes it is in the buyers best interest to write a personal letter to the seller, that their agent will pass along to the seller, discussing how the home fits the buyers families needs. The letter may bring an emotional response from the seller and "may" help swing the sale acceptance your way.
Also, it is no unusual for the seller to make a "multiple counter offer" back to both buyers to see if either were willing to make the terms a tiny bit better etc.
Sellers like fewer contingencies. However, possession dates and closing dates may vary depending on the sellers situation. Your agent should ask the listing agent questions about the sellers timeframes and if you can be flexible, do it.
Good luck on your offer
I hope this helps. Thanks for this great question.
When it comes down to it, most likely one will look more attractive to the seller than another. Sometimes the decision is based on their confidence to close (that it won't fall out of contract) and not just the price/terms.
Hope that helps.