Once a player makes the team he will most likely play ahead of a scholarship player if he can better help the team win. The coach has a direct need to win. If a players is a non-scholarship player and for whatever reason the coach views him as only equal or maybe slightly better than a scholarship player then if cuts are needed that Fall he will probably be let go. A scholarship player will be given more looks to prove he cannot play at a level needed. The coach has invested money in the player and would look like a very poor recruiter if he cuts scholarship players over non-scholarship players. However at the end of the season the coaches may "invite" a scholarship player to leave if he doesn't think he can contribute. My son had a former high school team mate that after his Sophomore season was informed he probably would not receive much playing time in the future and would be offered the opportunity to make the team. The players offensive stats and defensive play was not such that I would have thought he would be in that position, but it appeared after the fact the coach wanted to invest the scholarships in a rebuild of sorts with JUCO or Freshman players. The player moved to another program rather than face the somewhat uphill battle.
Academic money and athletic money can both be offered if appropriately deserved. I would be cautious in taking a non scholarship offer over a scholarship. When 45 players show up in the Fall chances are the scholarship players are going to make the team.