If converting modules, your PC's will advance more slowly at first
One of the campaigns I currently run is dedicated to taking first edition Greyhawk modules, converting them to 5e, and running a 5e Greyhawk campaign. The party started in U1 with 0xp and have done U2, U3, I7, C2, DF32, S3, G1, G2, and are currently in G3 having just leveled to 11th.
It is fundamental to understand that in BECMI and OD&D, treasure was the primary source of xp, and typically 80% or so of a character's advancement came from treasure recovered rather than monsters defeated. Thus, if you are going to take a old school module and convert it to 5e by simply using the 5e monster equivalents and reducing the treasure values by 10 (which is largely what I did; read the link if interested), any given encounter should play well enough, but over time you will quickly find that PC's are not advancing at the rate the module assumes they will. Some of this will be offset by smaller party sizes; BECMI and especially OD&D assume far larger party sizes with xp divided among more participants. But even a small, thorough 1st level party would struggle, for example, to find enough monster xp in the first half of U1 to level to 2nd by the second half of U1. The module assumes that they will be 2nd level for the second half, and thus such a party will find themselves outmatched by the power of their opponents. Thus, along with reducing the treasure values, in my game I also introduced story-based leveling, largely so that the party could keep pace with the challenges required without having to add numerous encounters and sidequests for them to accumulate monster-based xp.
If, as you presuppose, you are retaining the original treasure values but converting the module monsters and still only earning xp for monsters as standard for 5e, you will now still have slow xp advancement and leveling, but your characters will also be flush with cash, as Dan B points out. Because the PC's are advancing slowly, they will find the module challenges more and more difficult to face, particularly if they are playing in a module series. Because they have abundant treasure, they will likely attempt to use that treasure to increase their power to meet the challenges presented. What happens then depends on your campaign and what sort of markets are available to the PC's.
What happens next depends on you
If your world has a broad labor market, such that the PC's are able to use their wealth to hire NPC's, this will help them in the short term. Hiring mercenaries and guards is closer to how the original modules were typically played, and the PC's should find this a successful strategy at first. However, since those NPC's will also take a share of the xp (cf DMG 260), this will, in the long term, make PC progress even slower. Taken to its logical conclusion if trying to follow a module series progression, the PCs will eventually become the officers and owners of mercenary companies, as they do not individually have the power to face the challenges of the modules, but by having old-school treasure values in a 5e economy they can hire the small armies they need to succeed, thereby gaining more treasure to continue the process.
On the other hand, if your game has a tight labor market but a fluid market for magic items, the PC's will use their wealth to make themselves individually more powerful than their levels alone imply. If everything is available and they are astute about which items they purchase, they can find success with this strategy as well. However, combats may become much more 'swingy', as monsters with strategies the PC's are equipped to counter are shut down fast, but monsters for which the party's current inventory of magic items does not prepare them hit the PC's high above their actual level.