They're different tech

The important thing to recognise is that these are two vastly different pieces of technology, running on quite different hardware (one a Starfleet ship's computer and Hilo-emitters, the other a positronic brain). So there is a difference in programming them by virtue of their makeup.

Intentional design

The other factor and probably most important one is to remember that Data was made by Soong after Lore because Lore's emotions twisted. Data was designed and programmed intentionally to be emotionless because of Lore's problems.

Besides the answers already mentioned (Data being intentionally stripped of emotions and the Doctor being later technology), there's another aspect worth considering: that the Doctor does not truly have emotions.

Judging by the Lewis Zimmerman we see in in the episode "Life Line", the Doctor was modeled after an irritable, sarcastic man. The Doctor's complaints about the crew and other emotions may be no more than a simulation Zimmerman designed to replicate his own commentary. Most holographic characters display emotion, and the Doctor's may be as superficial as the melodramatic characters we see in Reginald Barclay's holoprograms.

Just to add a little more to the mix. Out of universe, Data having no emotions gave the writers and story a place to explore emotions. Several times in TNG Data displays, what may be considered as, simple emotions. He has preferences, and most importantly desire. He is "fascinated" quite often and even displays sadness, angst, and hope.

The trick is that he doesn't seem to know he is doing it. Sure Data's emotions may be underdeveloped, but in several Data story lines, I get the distinct sensation that Soong wanted Data to develop his emotions naturally instead of being given a prescribed set of emotion algorithms. In fact that was one of the lessons learned from Lore.

The doctor on the other hand was designed to be an emergency replacement, running for only a couple of hours in a crisis. Generally thought of as a massive failure, the MK1 EMH's emotions got in the way more often then not. (see DSN episode and Voyager's Lifeline episode) But the emotions were programmed in to make it easier to relate to patients.

Simply put, they both had "emotions" but they started from different places. The doctor's was programmed in, and Data's was left to be discovered.

The holograms have the capability to draw upon the entire resources of the mainframe of the ship. We see this in the TNG episode, Ship in a Bottle, where Moriarty becomes self-aware. The programming of the ship mainframes is complex enough that it is apparently capable of creating self-aware AI. Housing that in a mobile brain is a hardware problem.

In TNG's Measure of a Man, we learn that the Federation has not yet managed to recreate the unique hardware of Data. Data's emotion chip is a chip, a piece of hardware, not an update to his programming but instead an add-on.

So it comes down to this: The software is fine, it's the hardware that is the issue.

tl;dr: Apples and oranges.

There are several problems with this question.

First, we've never discovered for sure that the EMH does "feel emotion". Right up to the end of the show, several characters (despite having become his "friend") still considered his personality a mere simulation, and a hearing in late Season 7 declined to confidently declare that he had sentience. The only person to outright state this was the alien in Tinker, Tenor, Doctor, Spy and, at the time, he was wrong about the Doctor in many, many ways.

(Mind you, Star Trek has always been unclear and contradictory on the matter of whether its holograms could be self-aware; see e.g. Moriarty in early TNG. It's possible that this capability has been effectively retconned to a degree as holographic characters became a core part of the lore.)

Second, Data was eventually imbued with emotions (with tech that far predated the EMH), and it has always been made clear that Data is actually a sentient "lifeform". These emotions were assuredly "real".

I just don't think that the two can be compared in this manner.

A common explanation I don't see represented here is that Voyager's computer actually has biological circuitry, while Data is entirely mechanical. It is often proposed that the bio-neural gel packs are at least part of the reason that the EMH was so easily able to become more human in his sentience.

That said, there are other examples of sentient holograms (all created after Data) that seemed to have emotions. Moriarty seemed to have emotions, even though it took a lot of the ship's power to create him. Minuet seemed to have emotions when the Bynars temporarily upgraded the holodeck system in order to lure in Picard and Riker. Vic Fontain seems to have an emotional awareness in his counseling showing at least greater emotional awareness than Data. And the Fair Haven folk became self-aware and expressed the emotion of fear as their main plot.

Only the latter were made on a ship with bio-neural gel packs, which would seem to disprove this theory. But I note that the only other accidental formation of emotions took a lot of the ship's power to do so, so it's possible the bio-neural gel packs make emotions easier to form.

Anyways, this a common fan speculation to the point that it was originally included on Memory Alpha until removed for being non-canon. It was on so many sites that people assumed it must be canon.