The answer is… in the words themselves.

Vouloir is… a matter of… will!

When Aimer is a matter of… affect!

Will or affect ? Do you want or do you like ? What commands ? You say!

Apart from that take care of the mood. Conditional mood is used there. In French this is, in this précise case, what will express politeness, irrespective of the verbs.

A The difference can be rather intangible between wish and request.

There is a particularity worth considering in what concerns expressing a wish, insofar as it is a wish that entails someone’s submitting to what that wish is, and asking for something ; those two acts are different, however, when you express such a wish you do not ask explicitly for something and when you ask for something you understate the existence of a wish of yours, whereby an indissociable coexistence of the two.
When you say “j’aimerais” you are saying that you don’t have something (understated) and that you’d like to have it; when you say “je voudrais” you are also saying that you don’t have something (understated), but then there are two possibilities that the context determines sometimes sharply, and sometimes leaves indeterminate; in some cases the meaning is that of a wish; you are not then making a request from the person you are talking to. In other cases the meaning has to be that of a polite request.

If you’re at the butcher’s it’s only appropriate to say “Je voudrais deux steaks de veau bien tendres.” ; “J’aimerais …” will not do. Therefore, in the context of rightfully expected service there is no choice and no ambiguity: only “je voudrais” to express only a polite request.

If you are speaking about the evening meal with your wife, whether you say “J’aimerai que tu fasse du veau comme tu sais le faire.” or “Je voudrais…” can, according to the context, be nothing but the same expression of a wish;

• Je voudrais que tu fasse du veau comme tu sais le faire, mais puisque tu veux aller au restaurant ça sera pour une autre fois.; in this context “je voudrais” expresses a wish.

Nevertheless, in talking with your wife or speaking to a member of your household employees, in the way of a polite request both will do.

• Pour ce soir j’aimerais/je voudrais que tu prépares du veau comme tu sais le faire et un dessert glacé, nous avons un invité.
  That use, however, may reflect a certain condescending attitude towards your wife, an attitude we tend to consider as a standard of days gone by; that depends on the tone of the utterance. A plain question is always exempt of those connotations.

• Pour ce soir j’aimerais/je voudrais que vous prépariez du veau comme vous savez le faire et un dessert glacé, pour quatre.

When using an adverb with the verb “aimer” the conditional mood in using this verb is not even the unique means of formulating a wish or a request; the present can also be used.

For instance let’s consider the following sentences.

• J’aime(rais) autant que tu ailles au supermarché avec moi. → I would just as soon you went to the supermaket with me.
  The expression of a wish is just as valid as an interpretation as that of the expression of a request.

• J’aime mieux que vous fassiez cela tout seul! → I would rather you did that by yourself!
  This can be not too polite a request or a plain, rather curt injunction saying “Do that by yourself!”; it all depends on the relationship between the locutors (parent/child, employer/employee, friends,…) and on the intonation.

• J’aime(rais) mieux que vous ne disiez rien, c’est trop! → I would rather you said nothing, that’s too much!

All those translations are based on the modal verb “will” and that is essentially “vouloir”; we see then that “aimer” can express in French the idea of “vouloir” and that is confirmed by the English rendering. Let’s say as a remark on the side that in this situation of expressing wishes the traditional “Vos désirs sont des ordres.” can be the norm.

“Voudrais” can be used to express a simple wish too, but in a different context, not that, so far considered, of someone from whom something might be requested.

• Je voudrais vous voir plus sérieux. (I wish you were more serious.)

• Je voudrais (bien) avoir une télévision. (I wish I had a television.)

B According to what has been said, in the first sentence one can see the expression of a wish, basically, without excluding, on the count of the context at hand, undertones of a request (more or less polite), not to be confused with a demand whether or not the locutor will accept to do something, as then the utterance is rather the subdued statement of an order.