I'm going to answer myself here and say it's up to the DM.
This because it looks like they are indeed treated as 2 different entities in the Sword Coast Adventure Guide, but I guess that a DM can keep the 4e twist if he really liked the fact that they were the same deity with only different aspects.
Things might change when we have more books published, but this is the present situation as far as canon is concerned.
UPDATE: confirmed they are different entities again in 5e -> https://twitter.com/Sernett/status/877036497165139968
Print sources
There are lots of Forgotten Realms materials regarding deities, published over almost all editions of the game. For example, a lot of details are available in AD&D 2e sourcebook Faiths and Avatars (F&A), a 192-page tome dedicated only to the Faerunian pantheon. A tome of similar length is Faiths and Pantheons (F&P), this time for the 3e of the game. It is worth mentioning that the two books have somewhat different strengths though: F&A spends more space on clergy (day-to-day activities, holy days, ceremonies, priestly vestments - a color picture is given, adventuring garb), while F&P gives more details about the past mythology of the deities and their relationships with each other.
If you want to stick to 5e material, of course you also have the Sword Coast Adventurer's Guide, but the information will be relatively limited as deities are covered only as a chapter.
Novels
Lathander is a deity that is mentioned/featured in many FR novels. If you want to read about how their clergy or associated orders behave, you can try to have a look at this old list from candlekeep. (For example, it mentions that The Year of Rogue Dragon trilogy features Pavel Shemov, a cleric of Lathander.)
As mentioned in a comment by @Baergren, a more recent novel is The Reaver from the Sundering series, which discusses the "reemerging" of Lathander after a century-old disappearance.
Online sources
Regarding the Lathander/Amaunator issue, you can have a look at this post.
You can also read some information in the Forgotten Realms wikia.
Finally, another excellent website for finding all kinds of FR-related information is Candlekeep.com. It even gets posts from Ed Greenwood, the creator of the setting. If you use your favorite web search engine with the string "site:candlekeep.com lathander", you will find plenty of interesting things.
Best Answer
At various points in the history of the Realms, Amaunator and Lathander have been the same, separate, and/or non-existent. In particular, there was a lot of combination during the Spellplague1. Later, during the Second Sundering2 the overgod Ao rewrote the Tablets of Fate. Many old gods were restored to power, others were split into parts, and some were destroyed. There was a period of time when gods, via their "Chosen" had to collect mortal followers and restructure their churches.
The current state of affairs in the "modern" time period is that Amaunator and Lathander are two separate deities. Amaunator's teachings mostly center around light as a purifying power, a source and sign of justice and righteousness. Lathander is more about the dawn - new life, new growth. There's certainly some overlap in their portfolios, but they are different.
It's entirely possible that a given temple may give service to one or both deities. In smaller settlements, there are temples used with multiple deities - many of the Realms gods work together or are allied, so their faithful do the same (the triad of Tyr, Torm, and Ilmater is probably the most notable example). However, they are separate entities, which means there are also temples that only deal with one of them.
1The Spellplague was used as the justification for the drastically different mechanics behind magic in 4E.
2Returning spellcasting mechanics in 5E to a method more like 3.5E and before.