According to the guys who literally wrote the book on it, well, it depends on which one you ask.
Random dude: Can you jump farther than your movement when using magic i.e spell Jump & boots of striding and springing?
Mike Mearls: i'd rule yes - design intent is to make you jump super far
Jeremy Crawford: To be clear, things like the jump spell don't increase speed. You can jump crazy far, but your speed caps it.
From Facebook, "Mike Mearls led the creation of Dungeons & Dragons 5th edition. He oversees the game's development and leads the creation of D&D's story lines and worlds." While according to Twitter, Jeremy Crawford is "lead designer of the #DnD Player's Handbook, I answer rules questions here and compile answers monthly."
Together they're co-designers of the game, but Mearls tends to answer more along the lines of RAI or his house rules, while Crawford's answers are supposed to be the official RAW. I'm not entirely clear on whether random tweets are considered official, or just the Sage Advice Compendiums.
So, let's look at this from a few different perspectives.
From a physics perspective, there are three ways to jump farther.
1. You can increase your vertical speed (and therefore height) during the jump.
2. You can increase your horizontal speed during the jump.
3. You can decrease your falling rate after the jump.
The Jump spell says:
(a grasshopper’s hind leg) You touch a creature. The creature’s jump distance is tripled until the spell ends.
Of note is that the material element is a grasshopper's leg. This strongly implies the spell is boosting the recipient's strength while jumping. That is, the extra jump distance is very likely caused by 1 or 2 above, not 3.
Ok, so how do we increase jump distance through extra speed? Well, it's basic kinematics, but the gist is this:
py = ½ A t² + vy0 t
px = vx0 t
We solve the first equation to find out when vertical (y) position is 0, and we get t=0 (we started on the ground) and t=½Avy0. We can substitute this into the second equation to get:
d = vx0½ A v y0 = ½ A vx0 vy0
This means we can triple the jump distance by tripling either vertical or horizontal speed off the jump. If we triple the vertical speed, we also triple the time taken, and the horizontal speed is unchanged. If we triple the horizontal speed, we jump three times the distance in the same amount of time. We could multiply each by about 1.732 (sqrt of 3) so it would take 1.732 times longer.
So mathematically, we could go either way. However, there's a very strong argument for mostly horizontal speed. Jumping higher means your maximum height is increased by a factor of 9=3², and putting all that extra power into a vertical jump would let you jump even higher. A strength 10 character could jump (3+0)×9=27 feet in the air, while a strength 20 character could jump (3+5)×9=72 feet in the air. Or 2.5×9 = 22.5 feet high while jumping 30 feet forwards at strength 10 or 5×9 = 45 feet high while jumping 60 feet forwards at strength 20. Jumping faster means your maximum height doesn't change, and you just go a lot farther, which seems closer to the design intent here.
So RAW doesn't necessarily help a lot. Clearly, a normal jump is supposed to go about the same speed no matter how long the jump is. But that's because it's a lot easier to just use a fixed movement speed than do a bunch of math, and the approximation isn't that much different from reality. Plus, who's to say a super-long jump doesn't require some extra time to land and re-orient yourself, further mitigating the difference?
One thing that is helpful is this:
Long Jump. When you make a long jump, you cover a number of feet up to your Strength score if you move at least 10 feet on foot immediately before the jump.
Clearly, a long jump involves a running start. You aren't just moseying along; you're making a short dash. This further reinforces the notion that jumping a long distance is about speed, not height.
Using the numbers from the OP:
Normally, you'd use 10 movement before the jump, spend up to 10 points jumping (based on strength), then have 5 points to allocate before or after the jump. When the Jump spell, you're just tripling the distance you travel during whatever jump you take. So for each movement point you spend jumping, you travel 3 feet of horizontal distance.
RAW vs Physics
Something that really warrants pointing out about now: while the rules are clearly simplified more than the physics equations, they still map pretty nicely to real physics.
Using more physics, based on the numbers above. From Wikipedia, h = gt²/8. Solving for a height of 8 using 32 ft/s² gives about t=1.4 s on a strength 20 character. Jumping 20 feet in 1.4 seconds means horizontal speed is around 14 ft/s. Normal walking speed is 30 ft / 6s = 5 ft/s, so this isn't crazy. But the character isn't jumping vertically and horizontally at the same time.
From a forum: At your DM’s option, you must succeed on a DC 10 Strength (Athletics) check to clear a low obstacle (no taller than a quarter of the jump’s distance), such as a hedge or low wall.
So, while long jumping, your feet are a quarter of the jump distance off the ground, or about 5 feet on a 20-foot jump. This gives us only about 1.118 s of airtime. To clear 20 feet in 1.118 s, the character is moving about 17 ft/s horizontally or about 3.6 times normal walking speed.
A really fast sprint is about 45 km/h = 41 ft/s, and our guy gets up to around half that in 10 feet. In this article, we see the guy with the 41 ft/s speed, Usain Bolt, accelerates really hard off the line, reaching 5.5 m/s (18 ft/s) in about 1 s, over about 15 feet.
Similarly, the longest jump recorded was just shy of 30 feet, with more typical really long jumps being around 25 feet, unencumbered. So a 20-foot jump isn't crazy. Note that real jumpers typically have a greater acceleration distance, giving them take-off speeds between 12 and 35 ft/s, with normal ~20° jump angles being somewhere in the middle, but our 20-strength guy is accelerating quickly instead of smoothly. (Also, note that the 20° angle is yet again consistent with the notion that we're mostly looking for forward speed, not vertical height.)
Additionally, this is the kind of physics the game is intended to simulate as accurately as possible within the simplified rule systems. So "it's a fantasy world" isn't a good counter-argument in this case.
So using physics as a guide is a good start.
Ok, so physics plus some RAW plus the spell description says jumping with the Jump spell should mean you're traveling about three times as fast. Other people say you're traveling the same speed. How does that work in practice?
If we assume you travel faster while jumping farther under effects like this, everything seems to work out. You could try to cheat by walking 29 feet then jumping 60, but that doesn't work. You can only use up to your strength in movement points for jumping each round, and you can only use as many movement points as you have; you just travel three times further for each point used. So if you walk 29 feet then jump, you go 3 feet, for a total of 32. If you walk 20 feet then jump, you go up to 30 feet, for a total of 50 feet.
However, if we say your jump distance is limited by your speed, we have a problem. Say you're walking along and you see a ravine. It's 60 feet across. Your wizard buddy casts Jump on you, so you've got an entire minute to leap across. You walk up to the ravine, then in the last 10 feet you dash forward and leap 60 feet, easily clearing the ravine. Woot!
Then, you turn around and realize a goblin assassin is about to ambush your wizard friend! You yell at him so he's not surprised, then proceed to leap back across to help him. Except, you can't. You can't quite figure it out, but due to some meta-game thing called "we're in combat now", you can't go over 30 feet in one round, and 10 of those are the starting dash. So you can only get halfway across before falling in.
Ok, sure, you can use the dash ability to move even farther in a round. But out of combat, you didn't have to. And the jump spell is clearly an automatic dash action for the purpose of jumping. So you shouldn't have to. Plus, dash only gives you 60 feet, and the first 10 are used to start the jump, leaving you 10 feet short. So you have to take a full-round run action, when you should be able to leap across the ravine with your base move, use a dash action to add 30 feet of move and approach the goblin, then whack him in the face with your standard action.
Physics, real-world athletes, extrapolation from basic movement rules, and the spell description, tell us your movement speed should increase while jumping using the Jump spell. Your character's ability to run and jump inexplicably changes between non-combat and in-combat, which is just weird. And one of the game devs says you should be able to jump extra far, because that's the design intent.
So in your example, you could move 10 feet, use 10 of your remaining move points to quickly jump 30 feet, then move up to 5 more feet either before or after the jump, using up your 25 movement.
But, just to put a twist in everything, a different game dev (in this case, the one whose word is likely the official interpretation on RAW) says none of the above is true, and you're stuck at normal movement speed despite it making zero sense here. And the fact that there's no explicit RAW explaining how to deal with partial move distance means some DMs will be leery of calling that an official rule.
But I see it as an obvious interpretation of the RAW.