He still counts against the cap while on LTIR. His cap hit isn't removed. The cap ceiling is raised. The amount it is raised is dependent on their cap commitments at the time he is placed on LTIR.
The cap is set at 81.5 mil and Clarkson has a 5.25 mil cap hit. If Vegas were only at say 75 mil out of 81.5, putting Clarkson on LTIR would do nothing except open a roster spot, which regular IR already does. If they were at 80 out of 81.5, then their cap ceiling would go up to 85.25 mil. That's why teams try to get as close to the cap as possible when they have a player who will go onto LTIR to start the season. They want to max out their LTIR cap space.
The key difference is in how regular cap space and LTIR cap space is handled. Regular cap space can be banked. LTIR cap space can't.
The season is divided into days and the cap is divided by the total number of days in the season. Each team can spend up to that amount each day. So say for example the cap is 100 mil and there are 100 days in the season. Each team can spend 1 mil per day. Let's say Vegas is only spending 950k per day. That 50k that they aren't spending can be used later in the season. If they spend 950k for the first 50 days of the season, they can then spend up to 1.05 mil per day the rest of the way.
LTIR cap space doesn't work that way. Same situation, but they have a player on LTIR, raising their cap ceiling to 104 mil (1.04 mil per day). They are spending 1.02 mil per day. The 20k that they aren't spending each day is lost. They can't bank it and use it later. If they are spending 1.02 out of 1.04 per day for the first 50 days, they still can't spend more than 1.04 per day the rest of the way.
It's not an ideal situation to be in.