They are definitely not the same thing, either in Spanish or in English. In both languages the distinction is the same. Emergency is life-threatening; urgent needs prompt attention but is not life threatening. They are sometimes, though not always, dealt with in different places; there are urgent care centers in the US, which are not ERs. (Moreover, in the US they won't necessarily give you urgent care if you can't pay, but they'll always give you emergency care.)

"Qué es una urgencia y una emergencia
La atención urgente surge cuando se hace necesaria la asistencia inmediata [...]
Urgencia
Una urgencia se presenta en aquellas situaciones en las que se precisa atención inmediata.
Emergencia
Una emergencia es una situación crítica de peligro evidente para la vida del paciente y que requiere una actuación inmediata. [...]"
http://www.riojasalud.es/ciudadanos/centros-y-servicios/urge...

This page gives examples of each.

So it is in the UK, where "Urgent and Emergency Services" is a standard NHS term. And again, it's a question of life-threatening or not. For urgent care you call 111; for emergency, 999.

See the interim Keogh report here, especially e.g. p. 8:

"Firstly, for those people with urgent but non-life threatening needs we must provide highly responsive, effective and personalised services outside of hospital. These services should deliver care in or as close to people’s homes as possible, minimising disruption and inconvenience for patients and their families.
Secondly, for those people with more serious or life threatening emergency needs we should ensure they are treated in centres with the very best expertise and facilities, in order to maximise their chances of survival and a good recovery."
http://www.nhs.uk/NHSEngland/keogh-review/Documents/uecrevie...

See also:
http://www.nhs.uk/NHSEngland/keogh-review/Pages/urgent-and-e...

And for the US, this page is useful:

"Urgent care is not emergency care"
http://www.scripps.org/news_items/4231-should-you-go-to-the-...

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Note added at 3 hrs (2014-09-26 10:04:21 GMT)
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It may be urgent and emergency care or services depending on the specific context in each case.

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Note added at 8 hrs (2014-09-26 15:53:37 GMT)
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I don't want to over-extend what is already a long answer, but a couple more points:

First, to reinforce the point that "urgent" and "emergency" are clearly in principle distinct medical categories, here are the Wikipedia articles on emergency medicine and urgent care. The former says explicitly that: "Emergency medicine is distinct from urgent care, which refers to immediate healthcare for non-emergency medical issues." The articles are, as usual, primarily focused on the United States, but the distinction is internationally valid, as we have seen.
http://en.wikipedia.org/wiki/Emergency_medicine
http://en.wikipedia.org/wiki/Urgent_care

Second, a point about terminology. The question terms are about types of medical care and practice in which the nurses have worked, not the name of the hospital department in which they are delivered. In English, the types of care are called urgent and emergency care/services/medicine. Accident and Emergency (A&E) and Emergency Room (ER) are the hospital departments where emergency medicine is practised, and where urgent care is commonly provided also. But they are not the names of medical treatment facilities, not types of medical care, and are therefore not appropriate translations here. The approved generic term for this type of hospital department is now Emergency Department. Here, again, is the Wikipedia article:
http://en.wikipedia.org/wiki/Emergency_department

"Casualty" is an obsolete UK term (also used in some other countries) which is still commonly used informally but has no official status and should not appear in any formal medical-related document (such as a nurse's CV). In any case, once again, it is or was the name of a facility, not a type of medical care.

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Note added at 8 hrs (2014-09-26 15:55:57 GMT)
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Sorry: in paragraph 2 of the added note) I meant to say that A&E and ER are the names of medical treatment facilities; the word "not" before "the names" should not be there.