Many married couples have no will, and they don't necessarily need one, as long as they have decided to leave everything to their spouse, then split their assets equally among their surviving children after they both die. This path of inheritance is the normal distribution under many states' "intestacy statute" (the law setting forth the distribution of the property of a decedent with no will).
Unless a couple has multiple children and wishes to name a particular child as their executor (vs. one of the other children), there probably will not be any difference in who gets what, regardless of whether they created a will.
The Medicaid caveat
Once one spouse is enrolled in the Medicaid program, however, all of this changes.
For example, if the at-home spouse dies before the nursing home spouse, then all of the assets (bank accounts, personal property, real estate, auto, etc.) of the at-home spouse will pass into the name of the nursing home spouse. Since the Medicaid program limits the assets of a recipient of Medicaid benefits to $2,000, this most likely will immediately cause the nursing home spouse to become disqualified because they have too much money and are considered "over-resourced."
Note that in these situations, some (or even all) of the assets that come from the at-home spouse may be exempt under the Medicaid program. For instance, if the nursing home spouse used to reside in the home, then that house would be an exempt asset and not cause disqualification. The same rule applies to personal property and possibly even the couple's car. Money in the bank, however, would clearly be a problem, if it pushes the nursing home spouse above the $2,000 limit.
Even if all of the assets are exempt, this exposes the nursing home spouse to "recoupment" or "estate recovery" measures being taken upon their death. Under this law, the state must file a claim against the estate of a deceased Medicaid recipient who was at least age 55 or was in a nursing home at the time they received Medicaid. Since the assets are now titled in the name of the nursing home spouse (after the death of the at-home spouse with no will), the assets may have to be sold to repay the state.
How a new will can help protect assets
If the at-home spouse has a new will made, it may be possible to eliminate (or at least minimize) the ability of the state to recoup under the previously-discussed law. This process can get complicated.
It is not as easy as bypassing the nursing home spouse entirely and just leaving everything to the children because, under the laws of every state, a surviving spouse has the right to a certain portion of the estate of their deceased spouse, regardless of the terms of the will. In some states, this portion is one-half, in others it is one-third, and so on. In these situations, the state would treat the nursing home spouse as if she received that one-half (or one-third, as the case may be) and then gifted it to the persons who actually received the money under the will (usually the children). This gift is considered a disqualifying transfer, resulting in a penalty period that can be many months long, and during which Medicaid will not pay for the expenses of the nursing home spouse.
As you can see, there are many moving parts in this situation and you need an experienced Medicaid planning attorney to draft the at-home spouse's will, one who is aware of how to minimize the spouse's "statutory share" so as to leave as little as possible to the surviving spouse without causing Medicaid disqualification. In some cases—depending on state law—leaving the statutory share in a special needs trust for the surviving spouse may be the best solution. In any event, it is clear that some type of new will is always going to be needed once it is clear that one spouse will be receiving Medicaid for a nursing home stay.