No, they do not contain the same active ingredient. Benadryl (trade name) is also known as diphenhydramine (generic name); loratidine (generic name) is also known as Claritin (trade name). Both drugs are primarily antihistamines and primarily active antagonists at the H1 subtype of histamine receptor. It is this property that makes them effective for treatment of allergies.
The difference between the two medications is that diphenhydramine is a first generation or “non-selective” antihistamine. The term “non-selective” refers to these drugs' activity at a variety of receptors other than H1 - primarily, acetylcholine receptors. This can lead to more anticholinergic side effects such as dry mouth, urinary retention, etc. First generation antihistamines are also highly lipophilic, which means they cross the blood-brain barrier and have central nervous system activity. It is this property which makes them quite sedating (primarily due to activity at CNS histamine receptors).
Second generation antihistamines were designed to avoid some of the side effects of first generation agents. They are more specific for the H1 receptor and are mostly lipophobic, meaning they poorly penetrate the blood-brain barrier, resulting in fewer CNS effects. For more information about antihistamines, this Drug Class Review is quite informative.
Is one better than the other for hay fever?
Probably not. It’s just a matter of dose that determines how much anti-histamine activity each provides. The trouble is that many people will find the side effects of diphenhydramine and other non-selective agents to be dose-limiting, in which case practically it may not be as effective.
Should they be taken together (staggered or otherwise)?
I see no clear benefit to this. Again, their activity against hay fever is mediated by the same receptor. As such, they will just have additive effects. Taking twice as much of one or the other would similarly increase activity against hay fever and/or to contribute to untoward side effects. So it’s “OK” to take them together in the same way that it’s “OK” to take twice as much of either one.
In general, I see little reason to use diphenhydramine for allergic indications unless there is a desire for the sedative properties. Loratadine and other second generation agents have overtaken it in usefulness in this area.