You make no mention of any other applications used apart from gaming and then you make it clear that you prefer IPS panel technology to TN, despite the fact that - at your $250-$300 budget - all the IPS panel-based monitors will have response times around 5ms to 6ms. So I would assume that this preference is probably based on a combination of what you've read on sites like this one and your other (not mentioned so far) applications such as watching video clips/movies and some more mundane things like MS Office...
The two monitors listed in your post may well fit the budget parameters ($250-$300) have a number of common capabilities / features, but do differ in some key areas:
Common aspects:
* Diagonal Screen Size: ~24"
* Panel Technology used: A-IPS
* Colour Resolution: 16.7 million colours (i.e. 8-bit colour)
* Full panel adjustment options (Tilt/Swivel/Pivot/Height)
* Factory colour calibration
* HDMI and DisplayPort inputs
Differences:
The Asus PA248Q is an entry level member of Asus' ProArt range of (hold your breath) "Professional Artist" monitors designed to address the needs of such users. The panel technology used in this monitor is A-IPS (the entry-level IPS tech) and this panel - while being a 24" diagonal - has a very useful aspect ratio of 16:10 which, given the pixel density, results in a resolution of 1920x1200. The extra 120 pixels of vertical resolution might not sound significant but - particularly when working on documents or image that are "portrait-oriented" - do allow "readable" full-page views of these portrait files without having to pivot the monitor from landscape to portrait (and change display options).
The Dell U2414H has a resolution of 1920x1080 (i.e. 16:9 aspect ratio).
(There is another model from Dell which also features 1920x1200 resolution - Dell Ultrasharp U2415 - but this model is 10% over your upper budget parameter.)
The Asus also has some minor but useful capabilities such as selectable screen grids and page sizes.
Finally (and this is where you need to check locally) you need to consider and investigate the standard/level of after-sales support/service that is the norm in your area as this will depend more on the local area distributor than the brand. The best way to play this is pop in and ask the manager straight out questions like what are your policies regarding failed product - immediate replacement ex stock, return to manufacturer, etc. Once you've gathered all the facts, your call...
I hope that helps...
Dave
PS: I owned an Asus PA248Q for a couple of years before I started lusting after their PA279Q and arranged a mortgage to cover the - not insignificant - cost. At the time I wound up with three IPS monitors - PA279Q, PA248Q and an older LG 23" FHD monitor. A colleague started nagging me to sell him one and I only needed two, so sold him the PA248Q (higher price) and kept the LG as a "luggable" second monitor for my laptop (it was a flimsy device but worked). He still has the PA248Q and still raves about it (I smile smugly and say nothing about the PA279Q...

)
The only reason for moving was the increased size (27" vs 24") and the higher resolution (2560x1440) plus the move to an AH-IPS (top-of-the-range) panel technology - but the price premium was difficult to justify...