Without the cream of tartar I don't think it keeps as long. Though expect it will still last a week or so, if kept refrigerated.
Playdoh is so quick to make, though can be messy to get out of the bowl. One thing that can help is to apply some oil to the inside of the bowl before you start.
Don't use a saucepan, just use a big glass bowl and use hot water.
Children I babysit love making this, keep on asking me to make it... but their parents never have a big bag of salt - I'd better start keeping bags of salt in my car!