The are three differences between children and issue, in the sense that is relevant here (obviously, the latter word also has other senses).
(1) Issue is a broader term: it encompasses all the lineal descendants of a person, including the person's children, but also grandchildren, great-grandchildren, etc.
(2) Issue is, grammatically, a mass term. While one can say that somebody has five children, it is not normally said that somebody has five issues (if one insisted on using issue to convey that idea, one would have to say that the person's issue consists of five children).
(3) As a matter of style, issue is used only when discussing genealogy and the legal matters that concern succession and inheritance. It is likely to be understood only by those who have greater than average interest in such topics. Attempting to use it in any other context will result in puzzled looks, and quite often will simply not be understood.
In these limited contexts, issue is used in both American and British English.
Contrary to what the OP assumes and what one of the other answers implies, the meaning of issue has nothing to do with the position that those involved have in the society. In the relevant contexts, anybody's descendants can be, and are, referred to as the person's issue, regardless of the person's social status (although, obviously, if a person was rich or had a nobility title, more may depend on who that person's issue is, ).
To answer the other part of the OP's question, even in the contexts in which the use of issue is appropriate, it would be unusual to use it as a verb, and say that the parents issued their child. If somebody said that, it would be understood (by those who are familiar with issue as a noun, in the relevant sense), but perceived as odd. This is similar to the fact that, even though offspring is widely used as a noun, it would be odd to say that somebody sprung off his parents.
(This answer incorporates some of the points that have already been made by other contributors to this page, but attempts to consolidate them, and disentangle them from the remarks that may mislead the future visitors to this page.)