Wolfram Inc. is still a properitary company. So the first established rules govern all. A guide is the price for the product. That is a crude simplificator. So some prodcuts include others at all or partly. These facts make it more difficult to decide. Comparing is really hard and depends on the intentions.
Desktop for example includes Wolfram Pro to some extend. So the online solution Wolfram Alpha for free is the starting or an appetizer. Wolfram Alpha Pro is already up to Wolfram Mathematica Desktop for the smaller Editions. Desktop is not a real ad hoc criteria. It depends on the machine the software package is running on. It is not as easy as for example Jupyter local to for example kaggle.com at lower cost.
I suggest for an in themes discussion to use a search engine for the themes. For example differential equations. There are exhaustive benchmarker available for an independent consultation. Same is for integration. Onliner make the flaw decision not to the use the older and more in depth resources as integrals.wolfram.com. New users on Mathematica desktop do not rely on the knowledge representation of the built-in free form input and Wolfram Alpha Query. Knowledge is so important to the scienctific methodology I should not be underestimated.
Wolfram language is discussed in extent elsewhere an to long to be cover in depth here. But for most beginners that much to much. They are in need of a coach that covers their topics and is at hand when questions arise. So the explorations need guidance and that is neither in the online tools nor in the desktop version.
One advantage of the desktop version is unlimited input cells and usage time in the available resources. One limitation is the the online version may be in front or inmediate or backend be better monitiored. But that is only of real importance for intermediate levels. These are rapidly reachbed because the learning curve is steep for all Wolfram products.
Therefore a combination of all and gathering the best of all in probably the best option. Start at the optimal knowledge level well informed and with the right practice and experience makes the best options.
A nice example is calculating with physical dimensions. This is neither easy to use in Wolfram Alpha nor in Mathematica beyond a certain level of low inferetial work. This is decoration in the programming sense at with the foundation of Mathematica and Wolfram Language base. There arise new problems not in sight for starters. These are avoided sticking to Wolfram Alpha because is lacks the sophistication of Mathematica.
A rich topic is guidance in what is nice, good or scientific level visualisation. Neither of Wolfram Alpha or Mathematica are good advisers for those question concering visualisation. So it may be difficult to brigde the gap for more specialised software for example in statistics and simulation. The same is true for procedures, processes that base on feature and function that have to be programmed into Mathematica by hand. This may take plenty of time in Mathematica or Wolfram Alpha advanded products but just to input the data elsewhere.
It is signature to the products of Mathematica or Wolfram Alpha to arise a question of Your kind. A nice address may be the national resellers that face that on a profressional level. But that is heading the wrong direction. The reseller follow the open scene advise that is implicit in the product portfolio that the buyer has to choose from.
Nice starting point is by choice elliptical+integrals.
So many question so little answer. Mix all together as long as You can afford it or it is economically sensible. Even add up other package to enrich the experience like the Python ecosphere, add hardware to the package for more real world return of expierences. Market local on the particular application and with results or output.
Have some orientating perspectives:
Integrals:
Integrals
Integrals
Integrate
DoAnIntegral
Integration MathWorld
and many more.
Even this page belongs to it: Contibute references.
Overview:
This pictures sequence show a hard reduction. And for sure even Stephan Wolfram is very unhappy about that. There must be a priorization introduced that is left over to the person in operation of the Wolfram tool suite landscapes. You may prefer for example this institution NIST elliptical integrals or all the other coriphees list in that search engine result page (SERP). This even guides You to stackexchange.com. You will prefer to switch to this SERP: elliptical integrals good books and still need very much help.
But both can not compare to benchmark differential equation maple mathematica and beyond that may rest a very good answer to Your very good question. Some comparisions are accessable by benchmark cas maple mathematica. These are not to uptodate. These benchmarks have their flaws by design and the choice of perspectives they judge on.
A interesting perspective is requirements for certain task or commercial success. And with later Your question collapses since Mathematica and the succecessors are not commercially successful in the sense market leader in their niches. Computer Algebra Systems
or this review: 3Ms_for_Instruction_Reviews_of_Maple_Mathematica_and_Matlab. This type of reviews may be interesting for a decision Computer algebra system offer another standards oriented approach. But this too leave the space for judging objectively wide open. But is shows up some question to solve that is left out in the former reference review and it offer interesting references.
Back to the landscape from Wolfram. It is not always necessary to have the full power at hand. The smaller packages are interesting still up to universitary level if not to complex the targets of the solutions required.
I am better if the requirements are much tougher. I understand that the start is always more complicated and full of unknowns. There are indeed plenty of question closely related to this on mathematica.stackexchange.com. Some of these are really high rated. For example performance tuning in mathematica. That is something left out in the cheaper packages of the Wolfram versioning. But that requires more knowledge and skills that are for example only accessible via this community or others.
A local solution has the advantage of keeping the solutions better. Since many linux fans perfer text to other storage formats there is in both approaches some steps to be taken. While to online version content management systems are available but still rely on markup like pocket, there is need to reduce the fully worked through Mathematica notebooks to text or near text format. But only that a comfortable managed with CMS.
There are close to step-by-step guide available on the internet, like Wolfram Alpha that confirm my advise closely and fully.
Greatest problem that Wolfram left out is, how to manage your own notebooks or result pages from online requests. Many prefer enhanced text formats with markups like the question pages on this community. But it is still hard to retrieve information, useful information from that. That guides into the wide open terrain on what do this packages not offer. Some of these question are collected on this page where can i find examples of good mathematica programming practice. TiddlyDesktop is recommended because one can incluce MathML easily into it, so that the cell may be saved in normal html and via link executed in Mathematica.
This open source combination is the true present most wanted enhancer to Wolfram customers. Such page may be produced automated via Mathematica notebook interface. But one can use the documentation system of Mathematica itself. This leads because it is available online to the least cost solution. Do not but Mathematica but use the online documentation an run the questions You have in the free package and look how far You get and save everything in for example TiddlyDesktop for later usage.
The boundary are very tight but this may work further than a single package and provide You more convinience.
Buy the biggest Mathematica and Wolfram Alpha packages and reign the world it the other answer part. Short answer are hard because many intents and wishes may not be covered. This answer uses fundamental exploration with fundamental methodologies from so many different knowledge clusters it will be unbeaten for years and even by Wolfram Inc.
