Does anything different happen during General Conference when there's a new president?

Replacing the president of the Church is a special thing. I addressed the process of calling a new prophet in this answer. That process is different than what would be used to call, for example, a Ward Sunday School President. It's no surprise that the process of sustaining a new president of the Church is also different.

D&C 26:2 teaches that all things in the Church must be done with the common consent of the congregation. "Congregation" has a lot of different scopes in our church. For example, a Ward (our most common congregational unit) will sustain (vote to approve or accept pursuant to the Lord's command to act with common consent) a new Bishop. But no other Ward will vote to sustain that Ward's Bishop. The scope being only that Ward.

The President of the Church requires the sustaining vote of the entire congregation of the Church. When this is done the first time, it is done in a Solemn Assembly.

LDS Apostle David B. Haight taught:

A solemn assembly, as the name implies, denotes a sacred, sober, and reverent occasion when the Saints assemble under the direction of the First Presidency. Solemn assemblies are used for three purposes: the dedication of temples, special instruction to priesthood leaders, and sustaining a new President of the Church. (General Conference, October 1994)

As he states, we use Solemn Assemblies for three purposes:

1. The dedication of temples.
2. Special instruction to priesthood leaders.
3. Sustaining a new President of the Church.

As a side note, #2 is interesting and intentionally a bit vague. Assemblies for that purpose are most commonly to accept new scripture or direction for the Church. I would need to do some research to see if D&C 136, received by Brigham Young, was accepted in a Solemn Assembly. The early church was still creating the organization the Lord expected and some procedures weren't formally "locked in stone" for a number of years. However, sections 137 and 138 were accepted as scripture in Solemn Assembly in the early 1980s along with Official Declaration 1 and Official Declaration 2. It should be noted that there have been differences between the date a revelation or declaration was received or written and the date it was accepted as scripture by the congregation. The process is intentionally slow.

The difference between a normal General Conference and a Solemn Assembly is procedurally small. Rather than receive a single sustaining vote from the congregation as a whole (the normal process), each priesthood quorum votes in turn with the general membership voting last. Again, we turn to brother Haight.

There is a pattern to solemn assemblies that distinguishes them from other general Church meetings where we sustain officers of the Church. That pattern, which was established by the Prophet Joseph Smith, is that the priesthood quorums, commencing with the First Presidency, stand and manifest by the uplifted right hand their willingness to sustain the President of the Church as a prophet, seer, and revelator, and uphold him by their confidence, faith, and prayers. The priesthood quorums of the Church so manifest by their vote. Then the general body of all the Saints stand and signify their willingness to do the same.

TL/DR

So, yes, the meeting is different. A new prophet is sustained (accepted by Common Consent) for the first time in a Solemn Assembly. That sustaining is different in that each priesthood quorum votes, followed by a vote from the general assembly. It is this specific and detailed process of exercising Common Consent that makes the meeting a Solemn Assembly.

Edit: Though I know of no official statement that it is this way, it appears historically (and was confirmed the morning of Saturday, March 31, 2018) that the Saturday morning session of General Conference is declared a Solemn Assembly (when they are required) and the sustaining vote takes place then. At all other times, the sustaining vote takes place during the Saturday afternoon session of General Conference. A curious detail, but only important from a procedural point of view.