Yes there is a difference. No, for all practical purposes these clauses are all the same: it would be hard to argue that the landlord's intention; that pets are not allowed, is subject to misunderstanding.
As to the differences:
- Shall has a duty to
- Must required to. Virtually indistinguishable from shall but can be applied to an inanimate object on which a duty cannot be imposed.
- May has discretion/permission to
- Will expresses a future contingency
Typically, a court will not allow pedantic interpretations where the intention is as clear as these clauses, however, for the sake of argument.
- "Tenants may not have pets in the house" - they have discretion/permission not to have pets i.e. having pets is not mandatory.
- "Tenants shall not have pets in the house" - they have a duty to not have pets.
- "Tenants will not have pets in the house" - in the future, they can't have pets but they can have them now.
- "Tenants are not allowed pets in the house" - permission to have pets has been refused
- "No pets in the house" - what it says, however, there is no obligation on ... anyone to make this happen.
- "Tenants agree to not have pets in the house" - this would probably be interpreted as a warranty rather than a term of the contract. Breach of a warranty can lead to damages or orders for performance but not termination of the contract i.e. if they break their agreement they can probably be forced to get rid of the pets but can't be evicted.
