What I'm calling support vs advice is a fairly standard division. Both kinds of discussion start with one person talking about something that bothers them. In a support conversation, they're looking for empathy, and the other person shows that they care by listening, showing that they understand, and emotionally comforting. In a problem-solving conversation, they're looking for solutions, and the other person engages by making suggestions. This distinction is important to recognize mainly because it's easy for the second person to get confused about which type of conversation they're in, and engage in the wrong way; in particular, offering solutions when the other person wants support can be pretty bad, because a person who wants support can perceive advice as criticism. Here's the standard amusing youtube link.

I'm not sure what to call it; I think it's been discussed on LessWrong before, but I'm not finding it by searching. Other possible terms for it:

• care vs help;
• emotional support vs problem-solving;
• feeling-oriented conversation vs solution-oriented conversation.

LMK what the more standard terms for this are, if indeed there are any.

Anyway, I'm not writing this post to communicate the support-vs-advice model; rather, to name a third conversational mode which has some advantages of the other two.

I'll tentatively call this a problem-oriented conversation.

First, note that one of the main ways to show empathy is to show that you care about what's going on with the other person, and to demonstrate that you understand. To listen well, get curious.

Second, note that it's not advisable to propose solutions right off the bat. People come up with better solutions if they first discuss the problem in more detail. Hold off on proposing solutions. 

You can probably see where I'm going: in a "problem-oriented conversation", the second person responds with curiosity to the problem, asking questions instead of proposing solutions. They hold off proposing solutions, instead just trying to understand what's going on for the other person.

Consider:

• The initial framing of a problem is rarely the right one. Often the problem turns out to be totally different if you talk about it a little while.
• Your "obvious solutions" are probably just as obvious to the other person, if they're thinking about the problem the same way you are. Once both of you see the problem in the same frame, it's probably not necessary to propose actual solutions.
  • (Though, sometimes, proposing "obvious" solutions can be incredibly helpful; I'm not saying you should never do that! Just that you should consider holding off.)
• Even when I want solutions, it's often unhelpful for the other person to get too far into "debugging mode" before they understand the problem in as much detail as I do. Their suggestions won't deal with my real problem.

The problem-oriented mode is not without its own problems, however.

I've recently found myself in a few conversations where I try to take a problem-oriented stance (sometimes as 'transmitter' and sometimes as 'receiver'). Some issues:

• When the other person really wants advice, the problem-oriented stance can be baffling. I found myself in a position where my questions were being misinterpreted as advice. "I just don't see what you're proposing that I do." Sometimes people take it as an axiom that a line of questioning isn't helpful if it's not connected with a proposed solution. A problem-oriented conversation is then axiomatically 'unhelpful'.
• Similar, as a 'transmitter' seeking a problem-oriented discussion of some issue, I might get frustration from others because I'm not pursuing solution-oriented lines of thought. The conversation might get shut down as unfruitful before I've gotten the understanding I was looking for.
• Another problem is that problem-oriented conversations can get too meta and make a bigger problem out of things. For example, lots of specific household issues turn into "communication problems" if you keep looking for the larger root of the problem. You can end up in a conversational frame where it sounds like you need to have a big intervention. 
• More generally, things will go poorly if there's an implicit assumption that any aspect of the problem you talk about, you're saying needs to be changed. There needs to be room to explore the problem without necessarily suggesting action. Otherwise it isn't possible to explore things from all angles.

Still, this strikes me as an important mode of conversation.