Purchase train tickets online early or buy at station?

"Is it cumbersome to have to keep track of a ticket on your cell phone two weeks out?"

Ah, nope - I download the ticket pdf to my Documents App and rename with the date of travel. If you have an iPhone you can also download the ticket to your Apple Wallet. I do both, it's a built-in back-up.

As for line-ups - it's more about location and time of day.

"Is there a price incentive to buy two weeks out?"

Shorter trips NO, longer trips there may well be "substantial" savings. And sometimes, even if it's a short trip, I'll buy the ticket online the night before and go straight to the track in the morning.

Have a look at the Trenitalia and the Italo websites to see if it's cheaper to buy tickets in advance for the longer of your journeys.
If you do buy in advance, you are locked into that train at that time; so you might want to wait and buy when you are there, to be more flexible.

For all trains except Regionale (one fixed price - cheap), there are three levels of pricing --- Super Economy, Economy, and Base. The first two are discount pricing that has a limited number of tickets available for that train at that time. Once all of the limited number of tickets are sold they are no more. No additional discount tickets at a later date. The Base ticket price is always available unless the train is sold out -- rarely happens except under special situations such as holidays. The discount tickets come with lots of restrictions -- mostly no change, no refund. It is easy and generally quick to buy tickets are the station. Besides the tickets windows there will be banks of ticket issuing machines that are pretty easy to use. All tickets, except Regionale trains, come with a seat reservation and are specific to a train, date and time. If you miss a train with a discount ticket you need to buy a new one. If base fare ticket then you have about 90 minutes to get a new ticket for the ticket window. Some of the smaller stations will have only ticket machines.

I’d go ahead and purchase now if you know what days/ times you need.
Haven’t been to CT but with the reports of crowds I would assume long lines
It’s one less thing to do/ deal with.

If regional tickets you don’t have to validate the pre purchased tickets and they are good for that train or any other train later in day in case you miss it.

I don’t think you’ll find much of a discount for fast trains at this point anyway since you are traveling so soon.
If you don’t want to deal with showing tix on your phone you can print and bring hard copies ( I do this anyway just in case problems accessing on my phone)

Or get the easy to use app and purchase on that before getting to station

You might want to read up on trains in Italy on
Www.maninseat61.com

I found the ticket machines easy to use. Choose English as the language and it should be very straightforward. I haven't found lines to be long, but I have found them to be slow. (It's Italy, not Germany.)

You can go to https://www.trenitalia.com/en.html now and compare the fares for the trains you might want to take for your actual travel dates to the fares for travel tomorrow. That will give you an idea of whether you can save money by committing to specific trains now. You'll need to use the Italian station names. For Milan, used Milano Centrale. If you're also going to Florence, use Firenze S. M. Novella. You'll need to use one of the village names for your stop in the Cinque Terre. If you're going to Monterosso, that's the station to specify, not "Monterosso Marche".

There is no need to "compare the fares for the trains you might want to take for your actual travel dates to the fares for travel tomorrow"
Trenitalia has published the schedules of all trains up to June 12 and the full price tickets (Base) are always displayed in advance together with the discounted ones (Economy and SuperEconomy). Put your actual day of travel and compare the actual fares, you are not travelling tomorrow.

My wife and I will travel to Italy in 15 days.

On same direct IC trains running from Milano Centrale to Monterosso on May 20 there are SuperEconomy (€ 14.90 pp) and Economy tickets (€ 21.90 pp) left

Since the 20th is a Friday, there are no discounted tickets for sale on the IC departing in the morning. There are only Base tickets (€ 28.5 pp) left for the 8:05 IC train #657.

A difference of 7 € pp won't make a big difference; if you are actually travelling on a Friday and you want to depart from Milan at 8 AM, you'd better not wait the last minute to get your tickets.'

Most of The trains connecting the 5 Villages in the Cinque Terre are Regionale. Regionale tickets NOT purchased online are open, but you must time-stamp them before getting on the train.