For example, a Level 14 Elite Monster is worth 2000 XP.
A level 9 Solo is also worth 2000 XP.
Does this mean that these two monsters are of equal difficulty to an average party?
Here are some sample numbers (post-MM3):
These may vary depending on monster's role and flavor, but are a good baseline. As can be seen, and as can be predicted from the rules for adjusting monster levels, a level 14 monster has defenses and attacks that are 5 points higher. Post-MM3 these numbers don't depend on whether the monster is elite or solo.
The way 4e mathematics work, a party shouldn't be facing enemies that differ too much from them in level. That's its sweet spot, when players hit and get hit just often enough. Deviating from it is not fun, as it leads to one side "whiffing" all the time. In addition to that, higher-level monsters tend to have nastier effects, which a lower-level party may not be prepared to handle. As @BESW points out, this is particularly relevant when a monster is of a different tier than the party.
Finally, there is monster complexity to consider. A solo monster is designed to be the only monster in the encounter, to occupy most of GM's attention and provide most of the conditions for the battlefield. A level 14 party facing 2 level 9 solos would result in a slow and unnecessarily complex fight.
So while this may have been the original intent, to use encounter XP budget as the main criteria of encounter composition, there is more nuance to it than that. Instead, you should use tools provided to adjust monster level to within at most 2 of the party, reskin monsters to fit your needs, etc. - particular methods go beyond the scope of this question.
Bottom line: math may work out, but I highly discommend it.
No, XP cost alone doesn't mean the two monsters are of equal difficulty to an average party.
It comes down to mechanics beyond XP value. Solo monsters are theoretically* built to be able to handle multiple PCs at once, while elites are typically expected to have backup and thus do not have the range of powers which would allow them to stand alone against several PCs.
For example, solos typically have multiple action points, high save bonuses, and a number of immediate actions, all of which allow them to act more often (or prevent them from being locked down). Elites may have one action point, and/or a small save bonus, but typically the PCs will be able to quickly lock down the monster using such effects as daze, stun, slow, etc., and the battle then becomes one-sided.
Further, as Magician mentioned, enemies of a vastly different level than the party will be either impossible to hit, or impossible to miss. This, too, causes battles to become one-sided and either frustrating or boring to the players. In your example, players who would be fighting a level 9 solo would probably be levels 7-9, while players who would be fighting a level 14 elite would likely be levels 13-15. Consider the difference in PC attack bonuses and defenses between level 9 and level 15, and then consider how high they'd have to roll naturally to have a chance of hitting the elite's defenses, and how low it would need to roll naturally to miss theirs.
*[There is much disagreement over the effectiveness of solos in 4e, and a number of questions here on rpg.se discuss ways to fix the problems with solos.]
No. XP in no way equals a relative difficulty similarity in 4e.
Experience budgets, frankly, are just a rough (lousy) starting point in 4e and likely to overwhelm your party if you use them. I would recommend (regardless of prior RPG history), discarding the XP budget idea completely. After discarding budgets, I had fewer total party kills/legendary calkwalks and less adjusting to do under the radar when combat didn't go as planned.
Normally I would be tempted to chalk these changes up to being related to my own DMing style - but after speaking with other new and experienced DMs who mentioned this approach independently (and saw similar results), I am convinced that it is a worthwhile thing to do.
Also, remember budgets can inadvertently affect difficulty depending on party size, how well you use them, etc. But even when budgets work closer to how they are intended to operate, they are often far from exact...
Your focus should be on picking appropriate monsters your party can handle, with little focus on exact XP. 4e has a nasty habit of making players work really hard for tiny amounts of XP compared to the relative difficulty of the creatures that must be defeated. You will also minimize creature repetition - a problem if you haven't gathered a number of Monster Manuals, etc. to draw from. You may get stuck using the same creature(s) because they are the only ones near the parties experience level with the XP to fit your budget requirements.
Unfortunately, this leads to lack of appropriate XP to level in a timely manner (if that is part of your goal). If you want to run additional encounters above the recommended average of 10, feel free to do so. But realize that the majority of play time in your sessions will likely be involved with back-to-back combat/defeating traps, especially if your group meets infrequently.
If you are adding suitable levels of your own non-combat experience points to make sure your players are where you would like them to be, however, you should never have an issue with picking appropriate monsters for your parties abilities. Just figure out where you would like to have your players (or they would like to be) XP wise, and drop in the total amount of missing XP (e.g. not generated by monsters, etc.) into various goals in your campaign (finding items, helping people, accomplishing background goals, good role-playing, quick thinking during game play in combat or outside it, etc.)
Again, I would recommend AGAINST strictly adhering to the 4e guidelines for when you should dole out non-combat experience. They are extremely limited.
I have some other tips to recommend but I will give you (and anyone else who may want it) a TL;DR alternative to scanning the massive chunks of text below - for 4e, if you are short a player or two below 5, consider as a DM either running a tag-a-long NPC primarily to fill out the party or have one of the players do it. That way you will have less adjusting to do out of the box when trying to fit undermanned parties into the the 4e five character box.
Here some tips to remember when forming 4e encounters:
Remember that ALL your monsters involved in the encounter (Solo, Elite + friends, that unholy alliance of fairies and hell hounds, whatever) should be - in total - roughly equal to your players in basic stats and capabilities for a well-balanced encounter (not too hard, not too easy). As an example, if your have a five player party versus an equal level Solo plus two regular monsters (don't do this by the way), you are actually looking at the equivalent of roughly five characters vs. seven monsters (remember - Solos count for about five monsters in one based on stats. Elites are closer to two or three to one.)